Consistent Approximations for the Optimal Control of
Constrained Switched Systems---Part 1: A Conceptual
Algorithm
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Vasudevan, Ramanarayan, Humberto Gonzalez, Ruzena Bajcsy,
and S. Shankar Sastry. “Consistent Approximations for the
Optimal Control of Constrained Switched Systems---Part 1: A
Conceptual Algorithm.” SIAM J. Control Optim. 51, no. 6
(January 2013): 4463–4483. © 2013, Society for Industrial and
Applied Mathematics
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http://dx.doi.org/10.1137/120901490
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Society for Industrial and Applied Mathematics
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SIAM J. CONTROL OPTIM.
Vol. 51, No. 6, pp. 4463–4483
c 2013 Society for Industrial and Applied Mathematics
CONSISTENT APPROXIMATIONS FOR THE OPTIMAL CONTROL
OF CONSTRAINED SWITCHED SYSTEMS—PART 1:
A CONCEPTUAL ALGORITHM∗
RAMANARAYAN VASUDEVAN† , HUMBERTO GONZALEZ‡ , RUZENA BAJCSY§ , AND
S. SHANKAR SASTRY§
Abstract. Switched systems, or systems whose control parameters include a continuous-valued
input and a discrete-valued input which corresponds to the mode of the system that is active at
a particular instance in time, have shown to be highly effective in modeling a variety of physical
phenomena. Unfortunately, the construction of an optimal control algorithm for such systems has
proved difficult since it demands some form of optimal mode scheduling. In a pair of papers, we
construct a first order optimization algorithm to address this problem. Our approach, which we prove
in this paper converges to local minimizers of the constrained optimal control problem, first relaxes
the discrete-valued input, performs traditional optimal control, and then projects the constructed
relaxed discrete-valued input back to a pure discrete-valued input by employing an extension to the
classical chattering lemma that we formalize. In the second part of this pair of papers, we describe
how this conceptual algorithm can be recast in order to devise an implementable algorithm that
constructs a sequence of points by recursive application that converge to local minimizers of the
optimal control problem for switched systems.
Key words. optimal control, switched systems, chattering lemma, consistent approximations
AMS subject classifications.
49M25, 90C11, 90C30
Primary, 49J21, 49J30, 49J52, 49N25, 93C30; Secondary,
DOI. 10.1137/120901490
1. Introduction. Let Q and J be finite sets and U ⊂ Rm a compact, convex,
nonempty set. Let f : [0, 1] × Rn × U × Q → Rn be a vector field over Rn that defines
a switched system, i.e., for each i ∈ Q, the map (t, x, u) → f (t, x, u, i) is a classical
controlled vector field. Let {hj }j∈J be a collection of functions, hj : Rn → R for each
j ∈ J , and h0 : Rn → R. Given an initial condition x0 ∈ Rn , we are interested in
solving the following optimal control problem:
inf h0 x(1) | ẋ(t) = f t, x(t), u(t), d(t) , x(0) = x0 , hj x(t) ≤ 0,
(1.1)
x(t) ∈ Rn , u(t) ∈ U, d(t) ∈ Q, for a.e. t ∈ [0, 1] ∀j ∈ J .
The optimization problem defined in (1.1) is numerically challenging because the
function d : [0, 1] → Q maps to a finite set rendering the immediate application of
derivative-based algorithms impossible. This paper, which is the first in a two-part
series, presents a conceptual algorithm, Algorithm 4.1, to solve the problem described
in (1.1) and a theorem proving the convergence of the algorithm to points satisfying
a necessary condition for optimality, Theorem 5.16.
1.1. Related work. Switched systems, a particular class of hybrid systems,
have been used in a variety of modeling applications including automobiles and
∗ Received by the editors December 6, 2012; accepted for publication (in revised form) September
3, 2013; published electronically December 11, 2013. This material is based upon work supported by
the National Science Foundation under award ECCS-0931437.
http://www.siam.org/journals/sicon/51-6/90149.html
† CSAIL, Massachusetts Institute of Technology, Cambridge, MA 02139 (ramv@csail.mit.edu).
‡ ESE, Washington University in St. Louis, St. Louis, MO 63130 (hgonzale@ese.wustl.edu).
§ EECS, University of California at Berkeley, Berkeley, CA 94720 (bajcsy@eecs.berkeley.edu,
sastry@eecs.berkeley.edu).
4463
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4464
VASUDEVAN, GONZALEZ, BAJCSY, AND SASTRY
locomotives employing different gears [15, 26], manufacturing systems [8], and situations where a control module switches its attention among a number of subsystems [18, 33]. Given their utility, there has been considerable interest in devising
algorithms to perform optimal control for such systems. Since the discrete mode
of operation of switched systems is completely controlled, the determination of an
optimal control for such systems is challenging due to the combinatorial nature of
calculating an optimal discrete mode schedule. The theoretical underpinnings for the
optimal control of switched systems have been considered in various forms including a sufficient condition for optimality via quasi-variational inequalities by Branicky,
Borkar, and Mitter [7] and extensions of the classical maximum principle for hybrid
systems by Piccoli [21], Riedinger, Iung, and Kratz [25], and Sussmann [31].
Implementable algorithms for the optimal control of switched systems have been
pursued in numerous forms. Xu and Antsaklis [36] proposed one of the first such
algorithms, wherein a bilevel optimization scheme that fixes the mode schedule iterates between a low-level algorithm that optimizes the continuous input within each
visited mode and a high-level algorithm that optimizes the time spent within each
visited mode. The Gauss pseudospectral optimal control software developed by Rao
et al. [24] applies a variant of this approach. Shaikh and Caines [28] modified this
bilevel scheme by designing a low-level algorithm that simultaneously optimizes the
continuous input and the time spent within each mode for a fixed mode schedule by
employing the maximum principle and a high-level algorithm that modifies the mode
schedule by using the Hamming distance to compare nearby mode schedules. Xu and
Antsaklis [37] and Shaikh and Caines [29] each extended their approach to systems
with state-dependent switching. Alamir and Attia [1] proposed a scheme based on the
maximum principle. Considering the algorithm constructed in this paper, the most
relevant approach is the one described by Bengea and DeCarlo [4], which relaxes the
discrete-valued input into a continuous-valued input over which it can apply the maximum principle to perform optimal control and then uses the chattering lemma [5] to
approximate the optimal relaxation with another discrete-valued input. However, no
constructive method to generate such an approximation is included. Importantly, the
numerical implementation of optimal control algorithms that rely on the maximum
principle for nonlinear systems including switched systems is fundamentally restricted
due to their reliance on approximating needle variations with arbitrary precision as
explained in [20].
Several authors [9, 16] have focused on the optimization of autonomous switched
systems (i.e., switched systems without a continuous input) by fixing a mode sequence
and optimizing the amount of time spent within each mode using implementable
vector space variations. Axelsson et al. [3] extended this approach by employing a
variant of the bilevel optimization strategy proposed in [28]. In the low level they
optimized the amount of time spent within each mode using a first order algorithm,
and in the high level they modified the mode sequence by employing a single-mode
insertion technique. There have been two major extensions to this result. First, Wardi
and Egerstedt [35] extended the approach by constructing an algorithm that allowed
for the simultaneous insertion of several modes each for a nonzero amount of time.
This reformulation transformed the bilevel optimization strategy into a single step
optimization algorithm but required exact computation in order to prove convergence.
Wardi, Egerstedt, and Twu [34] addressed this deficiency by constructing an adaptiveprecision algorithm that balanced the requirement for precision with limitations on
computation. Second, Gonzalez et al. [12, 13], extended the bilevel optimization
approach to constrained nonautonomous switched systems. Though these insertion
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PART 1: A CONCEPTUAL ALGORITHM
4465
techniques avoid the computational expense of considering all possible mode schedules
during optimization, they restrict the possible modifications of the existing mode
schedule, which may terminate the execution in undesired stationary points.
1.2. Our contribution and organization. In this paper, we devise and implement a conceptual first order algorithm for the optimal control of constrained
nonlinear switched systems. In sections 2 and 3, we introduce the notation and assumptions used throughout the paper and formalize the optimization problem we
solve. Our approach, which is formulated in section 4, computes a step by treating
the discrete-valued input as a continuous-valued one and then uses a gradient descent technique on this continuous-valued input. Using an extension of the chattering
lemma which we develop, a discrete-valued approximation to the output of the gradient descent algorithm is constructed. In section 5, we prove that the sequence of
points generated by recursive application of our algorithm asymptotically satisfies a
necessary condition for optimality. The second part of this two-paper series [32] describes an implementable version of our conceptual algorithm and includes several
benchmarking examples to illustrate our approach.
2. Notation. We present a summary of the notation used throughout the paper.
·p
·i,p
·Lp
Q = {1, . . . , q}
J = {1, . . . , Nc }
U ⊂ Rm
Σqr
Σqp = {ei }i∈Q
Dr
Dp
U
(X , ·X )
Xr
Xp
N (ξ, ε)
Nw (ξ, ε)
di or [d]i
f : [0, 1] × Rn × U × Q → Rn
V : L2 ([0, 1], Rn ) → R
h 0 : Rn → R
{hj : Rn → R}j∈J
x(ξ) : [0, 1] → Rn
φt : Xr → Rn
J : Xr → R
Ψ : Xr → R
ψj,t : Xr → R
Pp
Pr
D
θ : Xp → (−∞, 0]
g : Xp → Xr
standard vector space p-norm
matrix induced p-norm
functional Lp -norm
set of discrete mode indices (section 1)
set of constraint function indices (section 1)
compact, convex, nonempty set (section 1)
q-simplex in Rq (equation (3.5))
corners of the q-simplex in Rq (equation (3.6))
relaxed discrete input space (section 3.2)
pure discrete input space (section 3.2)
continuous input space (section 3.2)
general optimization space and its norm (section 3.2)
relaxed optimization space (section 3.2)
pure optimization space (section 3.2)
ε-ball around ξ in the X -norm (Definition 4.1)
ε-ball around ξ in the weak topology (Definition 4.3)
ith coordinate of d : [0, 1] → Rq
switched system (section 1)
total variation operator (equation (3.3))
terminal cost function (section 3.3)
collection of constraint functions (section 3.3)
trajectory of the system given ξ (equation (3.8))
flow of the system at t ∈ [0, 1] (equation (3.9))
cost function (equation (3.10))
constraint function (equation (3.11))
component constraint functions (equation (3.12))
optimal control problem (equation (3.13))
relaxed optimal control problem (equation (4.1))
derivative operator (equation (4.2))
optimality function (equation (4.3))
descent direction function (equation (4.3))
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4466
VASUDEVAN, GONZALEZ, BAJCSY, AND SASTRY
ζ : Xp × Xr → R
μ : Xp → N
1A : [0, 1] → R
FN : Dr → Dr
PN : Dr → Dp
ρN : Xr → Xp
ν : Xp → N
Γ : Xp → Xp
Φ(ξ) : [0, 1] × [0, 1] → Rn×n
optimality function objective (equation (4.4))
step size function (equation (4.5))
indicator function of A ⊂ [0, 1], 1 = 1[0,1] (section 4.4)
Haar wavelet operator (equation (4.7))
pulse-width modulation operator (equation (4.8))
projection operator (equation (4.9))
frequency modulation function (equation (4.10))
algorithm recursion function (equation (4.11))
state transition matrix given ξ (equation (5.2))
3. Preliminaries. We begin by formulating the problem we solve in this paper.
3.1. Norms and functional spaces. We focus on the optimization of functions
with finite L2 -norm and bounded variation. Given n ∈ N, f : [0, 1] → Rn belongs to
L2 ([0, 1], Rn ) if
f L2 =
(3.1)
0
1
f (t)22
12
dt
< ∞.
Similarly, f : [0, 1] → Rn belongs to L∞ ([0, 1], Rn ) if
(3.2)
f L∞ = inf α ≥ 0 | f (x)2 ≤ α for almost every x ∈ [0, 1] < ∞.
Both definitions are formulated with respect to the Lebesgue measure on [0, 1]. In
order to define the space of functions of bounded variation, we first define the total
variation of a function. Given P , the set of all finite partitions of [0, 1], we define the
total variation of f : [0, 1] → Rn by
m−1
m
f (tj+1 ) − f (tj )1 | {tk }k=0 ∈ P, m ∈ N .
(3.3)
V(f ) = sup
j=0
Note that the total variation of f is a seminorm, i.e., it satisfies the triangle inequality
but does not separate points. f is of bounded variation if V(f ) < ∞, and we define
BV ([0, 1], Rn ) to be the set of all functions of bounded variation from [0, 1] to Rn .
The following result, which is an extension of Propositions X.1.3 and X.1.4 in [17] for
vector-valued functions, is critical to our analysis of functions of bounded variation.
Proposition 3.1. If f ∈ BV ([0, 1], Rn ) and v : [0, 1] → R is bounded and differentiable almost everywhere, then
1
f (t)v̇(t) dt ≤ vL∞ V(f ).
(3.4)
0
3.2. Optimization spaces. To formalize the optimal control problem, we define
three spaces: the pure discrete input space, Dp , the relaxed discrete input space, Dr ,
and the continuous input space, U. Let the q-simplex, Σqr , be defined as
q
q
q
(3.5)
Σr = (d1 , . . . , dq ) ∈ [0, 1] |
di = 1 ,
i=1
and let the corners of the q-simplex,
(3.6)
Σqp
=
Σqp ,
be defined as
q
(d1 , . . . , dq ) ∈ {0, 1} |
q
di = 1 .
i=1
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PART 1: A CONCEPTUAL ALGORITHM
4467
Note that Σqp ⊂ Σqr . Since there are exactly q elements in Σqp , we write Σqp =
{e1 , . . . , eq }. We employ the elements in Σqp to index the vector fields defined in
section 1, i.e.,
each i ∈ Q we write f (·, ·, ·, ei ) for f (·, ·, ·, i). Also note that for each
for
q
d ∈ Σqr , d = i=1 di ei .
Using this notation, we define the pure discrete input space, Dp = L2 ([0, 1], Σqp ) ∩
BV ([0, 1], Σqp ), the relaxed discrete input space, Dr = L2 ([0, 1], Σqr ) ∩ BV ([0, 1], Σqr ),
and the continuous input space, U = L2 ([0, 1], U ) ∩ BV ([0, 1], U ). Let the general
optimization space be X = L2 ([0, 1], Rm ) × L2 ([0, 1], Rq ) endowed with the norm
ξX = uL2 + dL2 for each ξ = (u, d) ∈ X . We combine U and Dp to define the
pure optimization space, Xp = U × Dp , and we endow it with the same norm as X .
Similarly, we combine U and Dr to define the relaxed optimization space, Xr =
U × Dr , and endow it with the X -norm too. Note that Xp ⊂ Xr ⊂ X . Note that we
emphasize the fact that our input spaces are subsets of L2 to motivate the appropriateness of the X -norm.
3.3. Trajectories, cost, constraint, and the optimal control problem.
Given ξ = (u, d) ∈ Xr , for convenience throughout the paper we let
q
f t, x(t), u(t), d(t) =
di (t)f t, x(t), u(t), ei .
(3.7)
i=1
Note that for points belonging to the pure optimization space, the bounded variation
assumption restricts the number of switches between modes to be finite.
We make the following assumptions about the dynamics.
Assumption 3.2. For each i ∈ Q and t ∈ [0, 1], the map (x, u) → f (t, x, u, ei )
is differentiable. Also, for each i ∈ Q the vector field and its partial derivatives are
Lipschitz continuous, i.e., there exists L > 0 such that given t1 , t2 ∈ [0, 1], x1 , x2 ∈ Rn ,
and u1 , u2 ∈ U ,
(1) f (t1 , x1 , u1 , ei ) − f (t2 , x2 , u2 , ei )2 ≤ L(|t1 − t2 | + x1 − x2 2 + u1 − u2 2 ),
∂f
(2) ∂f
∂x (t1 , x1 , u1 , ei )− ∂x (t2 , x2 , u2 , ei ) i,2 ≤ L(|t1 −t2 |+x1 −x2 2 +u1 −u2 2 ),
∂f
(3) (t1 , x1 , u1 , ei )− ∂f (t2 , x2 , u2 , ei ) ≤ L(|t1 −t2 |+x1 −x2 2 +u1 −u2 2 ).
∂u
∂u
i,2
Given x0 ∈ Rn , a trajectory of the system corresponding to ξ ∈ Xr is the solution to
(3.8)
ẋ(t) = f t, x(t), u(t), d(t)
for a.e. t ∈ [0, 1],
x(0) = x0 ,
and denote it by x(ξ) : [0, 1] → Rn to emphasize its dependence on ξ. We also define
the flow of the system, φt : Xr → Rn for each t ∈ [0, 1] as
(3.9)
φt (ξ) = x(ξ) (t).
The next result is a straightforward extension of the classical existence and
uniqueness theorem for nonlinear differential equations (see Lemma 5.6.3 and Proposition 5.6.5 in [22]).
Theorem 3.3. For each ξ = (u, d) ∈ X , differential equation (3.8) has a unique
solution.
(ξ) Moreover, there exists C > 0 such that for each ξ ∈ Xr and t ∈ [0, 1],
x (t) ≤ C.
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4468
VASUDEVAN, GONZALEZ, BAJCSY, AND SASTRY
To define the cost function, we assume that we are given a terminal cost function,
h0 : Rn → R. The cost function, J : Xr → R, for the optimal control problem is then
defined as
(3.10)
J(ξ) = h0 x(ξ) (1) .
If the problem formulation includes a running cost, then one can extend the existing
state vector by introducing a new state and modifying the cost function to evaluate
this new state at the final time, as shown in section 4.1.2 in [22]. Employing this
modification, observe that each mode of the switched system can have a different
running cost associated with it.
Given ξ ∈ Xr , the state x(ξ) is said to satisfy the constraint if hj x(ξ) (t) ≤ 0 for
each t ∈ [0, 1] and each j ∈ J . We compactly describe all the constraints by defining
the constraint function Ψ : Xr → R by
(3.11)
Ψ(ξ) = max hj x(ξ) (t) | t ∈ [0, 1], j ∈ J
since hj x(ξ) (t) ≤ 0 for each t and j if and only if Ψ(ξ) ≤ 0. To ensure clarity in the
ensuing analysis, it is useful to sometimes emphasize the dependence of hj x(ξ) (t)
on ξ. Therefore, we define component constraint functions ψj,t : Xr → R for each
t ∈ [0, 1] and j ∈ J as
(3.12)
ψj,t (ξ) = hj φt (ξ) .
With these definitions, we can state the optimal control problem, which is a restatement of the problem defined in (1.1):
(3.13)
inf {J(ξ) | Ψ(ξ) ≤ 0, ξ ∈ Xp } .
(Pp )
We introduce the following assumption to ensure the well-posedness of Pp .
Assumption 3.4. The functions h0 and {hj }j∈J , and their derivatives, are
Lipschitz continuous, i.e., there exists L > 0 such that given x1 , x2 ∈ Rn ,
(1) |h0 (x1 ) − h0 (x2 )| ≤L x1 − x2 2 ,
∂h0
0
(2) ∂h
∂x (x1 ) − ∂x (x2 ) 2 ≤ L x1 − x2 2 ,
(3) |hj (x1 ) − hj (x2 )| ≤ L x1 − x2 2 ,
∂h
∂h
(4) ∂xj (x1 ) − ∂xj (x2 )2 ≤ L x1 − x2 2 .
4. Optimization algorithm. In this section, we describe our optimization algorithm to solve Pp , which proceeds as follows: first, we treat a given pure discrete
input as a relaxed discrete input by allowing it to belong Dr ; second, we perform optimal control over the relaxed optimization space; and finally, we project the computed
relaxed input into a pure input. We begin with a brief digression to explain why
such a roundabout construction is required in order to devise a first order numerical
optimal control scheme to solve Pp defined in (3.13).
4.1. Directional derivatives. To appreciate why the construction of a numerical scheme to find the local minima of Pp defined in (3.13) is difficult, suppose
that the optimization in the problem took place over the relaxed optimization space
rather than the pure optimization space. The relaxed optimal control problem is then
defined as
(4.1)
inf {J(ξ) | Ψ(ξ) ≤ 0, ξ ∈ Xr } .
(Pr )
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PART 1: A CONCEPTUAL ALGORITHM
4469
The local minimizers of this problem are then defined as follows.
4.1. Let
an ε-ball in the X -norm around ξ be defined as N (ξ, ε) =
Definition
ξ¯ ∈ Xr | ξ − ξ̄ X < ε . A point ξ ∈ Xr is a local minimizer of Pr if Ψ(ξ) ≤ 0 and
ˆ ≥ J(ξ) for each ξ̂ ∈ N (ξ, ε) ∩ ξ̄ ∈ Xr | Ψ(ξ)
¯ ≤0 .
there exists ε > 0 such that J(ξ)
To concretize how to formulate an algorithm to find minimizers in Pr , we introduce
some additional notation. Given ξ ∈ Xr and any function G : Xr → Rn , the directional
derivative of G at ξ, denoted DG(ξ; ·) : X → Rn , is computed as
(4.2)
DG(ξ; ξ ) = lim
λ↓0
G(ξ + λξ ) − G(ξ)
λ
when the limit exists. Using the directional derivative DJ(ξ; ξ ), the first order approximation of the cost at ξ in the ξ ∈ X direction is J(ξ + λξ ) ≈ J(ξ) + λ DJ(ξ; ξ )
for λ sufficiently small. Similarly, using the directional derivative Dψj,t (ξ; ξ ) one can
construct a first order approximation of ψj,t at ξ in the ξ ∈ X direction.
It follows from the first order approximation of the cost that if DJ(ξ; ξ ) is negative, then it is possible to decrease the cost by moving in the ξ direction. In an
identical fashion, one can argue that if ξ is infeasible and Dψj,t (ξ; ξ ) is negative,
then it is possible to reduce the infeasibility of ψj,t by moving in the ξ direction.
Employing these observations, one can utilize the directional derivatives of the cost
and component constraints to construct both a descent direction and a necessary
condition for optimality for Pr . Unfortunately, trying to exploit the same strategy
for Pp is untenable, since it is unclear how to define directional derivatives for Dp
which is not a vector space. However, as we describe next, in an appropriate topology
over Xp there exists a way to exploit the directional derivatives of the cost and component constraints in Xr while reasoning about the optimality of points in Xp with
respect to Pp .
4.2. The weak topology on the optimization space and local minimizers. To motivate the type of required relationship, we begin by describing the chattering lemma.
Theorem 4.2 (Theorem 4.3 in [5]1 ). For each ξr ∈ Xr and ε > 0 there exists a
ξp ∈ Xp such that for each t ∈ [0, 1], φt (ξr ) − φt (ξp )2 ≤ ε.
Theorem 4.2 says that any trajectory produced by an element in Xr can be approximated arbitrarily well by a point in Xp . Unfortunately, the relaxed and pure
points prescribed as in Theorem 4.2 need not be near one another in the metric
induced by the X -norm. However, these points can be made arbitrarily close in a
different topology.
Definition 4.3. The weak topology on Xr induced by differential equation (3.8)
is the smallest topology on Xr such that the map ξ → x(ξ) is continuous with respect
to the L2 -norm.
Moreover,
an ε-ball in the weak topology around ξ is defined as
Nw (ξ, ε) = ξ¯ ∈ Xr | x(ξ) − x(ξ̄) L2 < ε .
More details about the weak topology of a space can be found in section 3.8
in [27]. To understand the relationship between N and Nw , observe that φt is Lipschitz
continuous for all t ∈ [0, 1], say, with constant L > 0 as is proved in Lemma 5.1. It
follows that for any ξ ∈ Xr and ε > 0, if ξ¯ ∈ N (ξ, ε), then ξ¯ ∈ Nw (ξ, Lε). However,
1 The theorem as proved in [5] is not immediately applicable to switched systems, but an extension
as proved in Theorem 1 in [4] makes that feasible. A particular version of this existence theorem can
also be found in Lemma 1 in [30].
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4470
VASUDEVAN, GONZALEZ, BAJCSY, AND SASTRY
the converse is not true in general. Since the weak topology, in contrast to the X -norm
induced topology, places points that generate nearby trajectories next to one another,
and using the fact that Xp ⊂ Xr , we can define the concept of local minimizer for Pp .
Definition 4.4. We say that ξ ∈ Xp is a local minimizer of Pp if Ψ(ξ) ≤ 0 and
ˆ ≥ J(ξ) for each ξˆ ∈ Nw (ξ, ε) ∩ ξ¯ ∈ Xp | Ψ(ξ)
¯ ≤0 .
there exists ε > 0 such that J(ξ)
With this definition of local minimizer, we can exploit Theorem 4.2, as an existence result, along with the notion of directional derivative over the relaxed optimization space to construct a necessary condition for optimality for Pp .
4.3. An optimality condition. Motivated by the approach in [22], we define
an optimality function, θ : Xp → (−∞, 0], which determines whether a given point is
a local minimizer of Pp , and a corresponding descent direction, g : Xp → Xr ,
(4.3)
θ(ξ) = min
ζ(ξ, ξ ) + V(ξ − ξ),
ξ ∈Xr
g(ξ) = arg min ζ(ξ, ξ ) + V(ξ − ξ),
ξ ∈Xr
where
(4.4)
⎧
⎪
⎪
max
max
Dψ
(ξ;
ξ
−
ξ)
+
γΨ(ξ),
DJ(ξ;
ξ
−
ξ)
⎪
j,t
⎪
⎪
(j,t)∈J ×[0,1]
⎪
⎪
⎪
⎨
+ ξ − ξX
ζ(ξ, ξ ) =
⎪
⎪
⎪
Dψ
(ξ;
ξ
−
ξ),
DJ(ξ;
ξ
−
ξ)
−
Ψ(ξ)
max
max
j,t
⎪
⎪
(j,t)∈J ×[0,1]
⎪
⎪
⎪
⎩
+ ξ − ξX
if Ψ(ξ) ≤ 0,
if Ψ(ξ) > 0,
with γ > 0 a design parameter, and where without loss of generality for each ξ =
(u, d) ∈ Xr , V(ξ) = V(u) + V(d). The well-posedness of θ(ξ) and g(ξ) for each
ξ ∈ Xp follows from Theorem 5.5. Note that θ(ξ) ≤ 0 for each ξ ∈ Xp , as we show
in Theorem 5.6. Also note that the directional derivatives in (4.4) consider directions
ξ − ξ in order to ensure that they belong to Xr . Our definition of ζ does not include
the total variation term since it does not satisfy the same properties as the directional
derivatives or the X -norm, such as Lipschitz continuity (as shown in Lemma 5.4).
In fact, the total variation term is added in order to make sure that g(ξ) − ξ is of
bounded variation.
To understand how the optimality function behaves, consider several cases. First,
if θ(ξ) < 0 and Ψ(ξ) = 0, then there exists a ξ ∈ Xr such that both DJ(ξ; ξ − ξ)
and Dψj,t (ξ; ξ − ξ) are negative for all j ∈ J and t ∈ [0, 1]; thus, using the first order
approximation argument in section 4.1 and Theorem 4.2, ξ is not a local minimizer
of Pp . Second, if θ(ξ) < 0 and Ψ(ξ) < 0, an identical argument shows that ξ is
not a local minimizer of Pp . Finally, if θ(ξ) < 0 and Ψ(ξ) > 0, then there exists a
ξ ∈ Xr such that Dψj,t (ξ; ξ − ξ) is negative for all j ∈ J and t ∈ [0, 1]. It is clear
that ξ is not a local minimizer of Pp since Ψ(ξ) > 0, but it also follows by the first
order approximation argument in section 4.1 and Theorem 4.2 that it is possible to
locally reduce the infeasibility of ξ. In this case, the addition of the DJ term in ζ is
a heuristic to ensure that the reduction in infeasibility does not come at the price of
an undue increase in the cost.
These observations are formalized in Theorem 5.6, where we prove that if ξ is a
local minimizer of Pp as in Definition 4.4, then θ(ξ) = 0, i.e., θ(ξ) = 0 is a necessary
condition for the optimality of ξ. Recall that in first order algorithms for finitedimensional optimization the lack of negative directional derivatives at a point is a
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PART 1: A CONCEPTUAL ALGORITHM
4471
necessary condition for optimality of a point (for example, see Corollary 1.1.3 in [22]).
Intuitively, θ is the infinite-dimensional analogue for Pp of the directional derivative
for finite-dimensional optimization. Hence, if θ(ξ) = 0 we say that ξ satisfies the
optimality condition.
4.4. Choosing a step size and projecting the relaxed discrete input.
Theorem 4.2 is unable to describe how to exploit the descent direction, g(ξ), since its
proof provides no means to construct a pure input that approximates the behavior
of a relaxed input while controlling the quality of the approximation. We extend
Theorem 4.2 by devising a scheme that allows for the development of a numerical
optimal control algorithm for Pp that first performs optimal control over the relaxed
optimization space and then projects the computed relaxed control into a pure control.
The argument that minimizes ζ is a “direction” along which to move the inputs in
order to reduce the cost in the relaxed optimization space, but first we require an
algorithm to choose a step size. We employ a line search algorithm similar to the
traditional Armijo algorithm used in finite-dimensional optimization in order to choose
a step size [2]. Let α, β ∈ (0, 1); then the step size for ξ is chosen as β μ(ξ) , where the
step size function, μ : Xp → N, is defined as
⎧
min k ∈ N | J ξ + β k (g(ξ) − ξ) − J(ξ) ≤ αβ k θ(ξ),
⎪
⎪
⎨
(4.5) μ(ξ) =
Ψ ξ + β k (g(ξ) − ξ) ≤ αβ k θ(ξ)
if Ψ(ξ) ≤ 0,
⎪
⎪
⎩
k
k
if Ψ(ξ) > 0.
min k ∈ N | Ψ ξ + β (g(ξ) − ξ) − Ψ(ξ) ≤ αβ θ(ξ)
Lemma 5.13 proves that if θ(ξ) < 0 for some ξ ∈ Xp , then μ(ξ) < ∞, which means
that we can construct a point ξ + β μ(ξ) (g(ξ) − ξ) ∈ Xr that produces a reduction
in the cost (if ξ is feasible) or a reduction in the infeasibility (if ξ is infeasible).
Now, we define a projection that takes this constructed point in Xr to a point
belonging to Xp while controlling the quality of approximation in two steps. First, let
λ : R → R be the Haar wavelet (section 7.2.2 in [19]):
⎧
1
⎪
⎪
1
if
t
∈
0,
,
⎪
⎪
2
⎪
⎨
λ(t) =
(4.6)
1
⎪
,1 ,
−1 if t ∈
⎪
⎪
2
⎪
⎪
⎩
0 otherwise.
Let 1A : [0, 1] → R be the indicator function of A ⊂ [0, 1], and let 1 = 1[0,1] . Given
k ∈ N and j ∈ 0, . . . , 2k − 1 , let bk,j : [0, 1] → R be defined by bk,j (t) = λ(2k t − j).
The Haar wavelet operator, FN : Dr → Dr , computing the N th partial sum Haar
wavelet approximation is defined as
(4.7)
k
N 2
−1
bk,j (t)
FN (d) i (t) = di , 1 +
di , bk,j 2 .
bk,j L2
k=0 j=0
The inner product here is the traditional inner product in L2 . Without loss of generality we apply FN to continuous inputs in U, noting that the only change is the domain
of the operator. Lemma 5.7 proves that for each N ∈ N and d ∈ Dr , FN (d) ∈ Dr .
We can project the output of FN (d), which is piecewise constant, to a pure
discrete input by employing the pulse-width modulation operator, PN : Dr → Dp ,
which computes the pulse-width modulation of its argument with frequency 2−N :
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1 if t ∈ SN,k,i−1 , SN,k,i , k ∈ 0, 1, . . . , 2N − 1 ,
(4.8)
0 otherwise,
i
where SN,k,i = 2−N k + j=1 dj 2kN . Lemma 5.7 also proves that for each N ∈ N
and d ∈ Dr , PN FN (d) ∈ Dp , as desired. Given N ∈ N, we compose the two
operators and define the projection operator, ρN : Xr → Xp , as
(4.9)
ρN (u, d) = FN (u), PN FN (d) .
PN (d) i (t) =
As shown in Theorem 5.10, this projection allows us to extend Theorem 4.2 by
providing a rate of convergence for the approximation error between ξr ∈ Xr and
ρN (ξr ) ∈ Xp . In our algorithm, we control this approximation error by choosing the
frequency at which we perform the pulse-width
modulation with a rule inspired in the
Armijo algorithm. Let ᾱ ∈ (0, ∞), β̄ ∈ √12 , 1 , and ω ∈ (0, 1); then we modulate a
point ξ with frequency 2−ν(ξ) , where the frequency modulation function, ν : Xp → N,
is defined as
(4.10)
⎧
min k ∈ N |ξ = ξ + β μ(ξ) (g(ξ) − ξ),
⎪
⎪
⎪
⎪
⎪
⎪
J ρk (ξ ) − J(ξ) ≤ αβ μ(ξ) − ᾱβ̄ k θ(ξ),
⎪
⎪
⎪
⎪
⎪
⎪
Ψ ρk (ξ ) ≤ 0,
⎪
⎨
ᾱβ̄ k ≤ (1 − ω)αβ μ(ξ)
if Ψ(ξ) ≤ 0,
ν(ξ) =
⎪
⎪
⎪
μ(ξ)
⎪
min k ∈ N |ξ = ξ + β
(g(ξ) − ξ),
⎪
⎪
⎪
⎪
⎪
⎪
Ψ ρk (ξ ) − Ψ(ξ) ≤ αβ μ(ξ) − ᾱβ̄ k θ(ξ),
⎪
⎪
⎪
⎩
ᾱβ̄ k ≤ (1 − ω)αβ μ(ξ)
if Ψ(ξ) > 0.
Lemma 5.14 proves that if θ(ξ) < 0 for some
ξ ∈ Xp , then ν(ξ)
< ∞, which implies
that we can construct a new point ρν(ξ) ξ + β μ(ξ) (g(ξ) − ξ) ∈ Xp that produces
a reduction in the cost (if ξ is feasible) or a reduction in the infeasibility (if ξ is
infeasible).
4.5. Switched system optimal control algorithm. Algorithm 4.1 describes
our numerical method to solve Pp . For analysis purposes, we define the algorithm
recursion function, Γ : Xp → Xp , by
(4.11)
Γ(ξ) = ρν(ξ) ξ + β μ(ξ) (g(ξ) − ξ) .
We say {ξj }j∈N is a sequence generated by Algorithm 4.1 if ξj+1 = Γ(ξj ) for each j ∈ N
such that θ(ξj ) = 0, and ξj+1 = ξj otherwise. We prove several important properties
about the sequence generated by Algorithm 4.1. First, Lemma 5.15 proves that if
there exists i0 ∈ N such that Ψ(ξi0 ) ≤ 0, then Ψ(ξi ) ≤ 0 for each i ≥ i0 . Second,
Theorem 5.16 proves that limj→∞ θ(ξj ) = 0, i.e., Algorithm 4.1 converges to a point
that satisfies the optimality condition.
5. Algorithm analysis. In this section, we derive the various components of
Algorithm 4.1 and prove that Algorithm 4.1 converges to a point that satisfies our
optimality condition. Our argument proceeds as follows: first, we construct the components of the optimality function and prove that these components satisfy various
properties that ensure the well-posedness of the optimality function; second, we prove
that we can control the quality of approximation between the trajectories generated
by a relaxed discrete input and its projection by ρN as a function of N ; and finally,
we prove the convergence of our algorithm.
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PART 1: A CONCEPTUAL ALGORITHM
4473
Algorithm 4.1. Conceptual optimization algorithm for Pp .
Require: ξ0 ∈ Xp , α, β, ω ∈ (0, 1), ᾱ, γ ∈ (0, ∞), β̄ ∈ √12 , 1 .
1: Set j = 0.
2: loop
3:
Compute θ(ξj ), g(ξj ), μ(ξj ), and ν(ξj ).
4:
if θ(ξj ) = 0 then
5:
return ξj .
6:
end if
7:
Set ξj+1 = ρν(ξj ) ξj + β μ(ξj ) (g(ξj ) − ξj ) .
8:
Replace j by j + 1.
9: end loop
5.1. Derivation of algorithm terms. We begin by noting the Lipschitz continuity of the solution to differential equation (3.8) with respect to ξ. The argument
is a simple extension of Lemma 5.6.7 in [22], and therefore we omit the details.
Lemma 5.1. There exists a constant L > 0 such that for each ξ1 , ξ2 ∈ Xr and
t ∈ [0, 1], φt (ξ1 ) − φt (ξ2 )2 ≤ L ξ1 − ξ2 X .
Next, we formally derive the components of the optimality function. This result
could be proved using an argument similar to Theorem 4.1 in [6], but we include
the proof for the sake of thoroughness and in order to draw comparisons with its
corresponding argument, Lemma 4.4, in the second of this pair of papers.
Lemma 5.2. Let ξ = (u, d) ∈ Xr and ξ = (u , d ) ∈ X . Then the directional
derivative of φt , as defined in (4.2), is given by
(5.1)
Dφt (ξ; ξ ) =
t
Φ
0
(ξ)
∂f τ, φτ (ξ), u(τ ), d(τ ) u (τ ) + f τ, φτ (ξ), u(τ ), d (τ ) dτ,
(t, τ )
∂u
where Φ(ξ) (t, τ ) is the unique solution of the following matrix differential equation:
∂f ∂Φ
(t, τ ) =
t, φt (ξ), u(t), d(t) Φ(t, τ ) ∀t ∈ [0, 1], Φ(τ, τ ) = I.
(5.2)
∂t
∂x
Proof. For notational convenience, let x(λ) = x(ξ+λξ ) , u(λ) = u + λu , and
d
= d + λd . Then, if we define Δx(λ) = x(λ) − x(ξ) , and using the mean value
theorem,
t q
(λ)
(5.3) Δx (t) =
λ
di (τ )f τ, x(λ) (τ ), u(λ) (τ ), ei dτ
(λ)
0
i=1
t
∂f τ, x(ξ) (τ ) + νx (τ )Δx(λ) (τ ), u(λ) (τ ), d(τ ) Δx(λ) (τ )dτ
∂x
0
t
∂f τ, x(ξ) (τ ), u(τ ) + νu (τ )λu (τ ), d(τ ) u (τ )dτ,
λ
+
∂u
0
+
where νu , νx : [0, t] → [0, 1]. Let z(t) be the unique solution of the following differential
equation:
∂f ∂f ż(τ ) =
(5.4)
τ, x(ξ) (τ ), u(τ ), d(τ ) z(τ ) +
τ, x(ξ) (τ ), u(τ ), d(τ ) u (τ )
∂x
∂u
q
+
di (τ )f τ, x(ξ) (τ ), u(τ ), ei , τ ∈ [0, t], z(0) = 0.
i=1
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We want to show that limλ↓0 λ1 Δx(λ) (t) − z(t)2 = 0. To prove this, consider the
inequalities that follow from condition (2) in Assumption 3.2,
(5.5)
t ∂f τ, x(ξ) (τ ), u(τ ), d(τ ) z(τ )
0 ∂x
(λ)
(τ
)
∂f Δx
τ, x(ξ) (τ ) + νx (τ )Δx(λ) (τ ), u(λ) (τ ), d(τ )
dτ −
∂x
λ
2
t
t
(λ)
Δx (τ ) Δx(λ) (τ ) + λ u (τ ) z(t) dτ,
≤L
dτ + L
z(τ ) −
2
2
2
λ
0 0
2
from condition (3) in Assumption 3.2,
t ∂f τ, x(ξ) (τ ), u(τ ), d(τ )
(5.6)
0 ∂u
∂f τ, x(ξ) (τ ), u(τ ) + νu (τ )λu (τ ), d(τ ) u (τ )dτ −
∂u
2
t
t
2
≤L
λ νu (τ )u (τ )2 u (τ )2 dτ ≤ L
λ u (τ )2 dτ,
0
0
and from condition (1) in Assumption 3.2,
q
t
(ξ)
(λ)
(λ)
(5.7)
di (τ ) f τ, x (τ ), u(τ ), ei r − f τ, x (τ ), u (τ ), ei dτ 0
i=1
2
t
q
≤L
di (τ ) Δx(λ) (τ )2 + λ u (τ )2 dτ.
0 i=1
Then, using the Bellman–Gronwall inequality,
(5.8)
t
Δx(λ) (t)
2
Lt
− z(t) ≤ e L
Δx(λ) (τ )2 + λ u (τ )2 z(t)2 + λ u (τ )2
λ
0
2
q
(λ) +
di (τ ) Δx (τ ) + λ u (τ ) dτ ,
2
2
i=1
but note that every term in the integral above is bounded, and Δx(λ) (τ ) → 0 for
each τ ∈ [0, t] since x(λ) → x(ξ) uniformly as follows from Lemma 5.1. Thus, by the
dominated convergence theorem and by noting that Dφt (ξ; ξ ) is exactly the solution
of differential equation (5.4) we get
(5.9)
Δx(λ) (t)
1
− z(t)
= lim x(ξ+λξ ) (t) − x(ξ) (t) − Dφt (ξ; λξ )2 = 0.
lim λ↓0
λ↓0 λ
λ
2
The result of the lemma then follows.
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PART 1: A CONCEPTUAL ALGORITHM
4475
We now construct the directional derivative of the cost and component constraint
functions, which follows from the chain rule and Lemma 5.2.
Lemma 5.3. Let ξ ∈ Xr , ξ ∈ X , j ∈ J , and t ∈ [0, 1]. The directional derivatives
of the cost J and the component constraint ψj,t in the ξ direction are
(5.10)
DJ(ξ; ξ ) =
∂h0 φ1 (ξ) Dφ1 (ξ; ξ )
∂x
and
Dψj,t (ξ; ξ ) =
∂hj φt (ξ) Dφt (ξ; ξ ).
∂x
Lemma 5.4. There exists an L > 0 such that for each ξ1 , ξ2 ∈ Xr , ξ ∈ X , and
t ∈ [0, 1],
(1) Dφt (ξ1 ; ξ ) − Dφt (ξ2 ; ξ )2 ≤ L ξ1 − ξ2 X ξ X ,
(2) |DJ(ξ1 ; ξ ) − DJ(ξ2 ; ξ )| ≤ L ξ1 − ξ2 X ξ X ,
(3) |Dψj,t (ξ1 ; ξ ) − Dψj,t (ξ2 ; ξ )| ≤ L ξ1 − ξ2 X ξ X , and
(4) |ζ(ξ1 , ξ ) − ζ(ξ2 , ξ )| ≤ L ξ1 − ξ2 X .
Proof. Condition (1) follows after proving the boundedness of the vector fields,
the Lipschitz continuity of the vector fields and their derivatives, and the boundedness
and Lipschitz continuity of the state transition matrix. Conditions (2) and (3) follow
immediately from condition (1).
k
k
To prove condition (4), first notice that for {xi }i=1 , {yi }i=1 ⊂ R,
(5.11)
max xi − max yi ≤ max |xi − yi | .
i∈{1,...,k}
i∈{1,...,k}
i∈{1,...,k}
Let Ψ+ (ξ) = max{0, Ψ(ξ)} and Ψ− (ξ) = max{0, −Ψ(ξ)}. Then,
(5.12)
ζ(ξ1 , ξ ) − ζ(ξ2 , ξ ) ≤ max DJ(ξ1 ; ξ − ξ1 ) − DJ(ξ2 ; ξ − ξ2 ) + Ψ+ (ξ2 ) − Ψ+ (ξ1 ),
max Dψj,t (ξ1 ; ξ − ξ1 ) − Dψj,t (ξ2 ; ξ − ξ2 )
j∈J , t∈[0,1]
+ γ Ψ− (ξ2 ) − Ψ− (ξ1 )
+ ξ − ξ1 X − ξ − ξ2 X .
Now, note that ξ − ξ1 X − ξ − ξ2 X ≤ ξ1 − ξ2 X . Also,
DJ(ξ1 ; ξ − ξ1 ) − DJ(ξ2 ; ξ − ξ2 ) ≤ DJ(ξ1 ; ξ − ξ1 ) − DJ(ξ2 ; ξ − ξ1 )
(5.13)
∂h0 + φ1 (ξ2 ) Dφ1 (ξ2 ; ξ2 − ξ1 )
∂x
≤ L ξ1 − ξ2 X ,
where we used the linearity of DJ(ξ; ·), condition (2), the fact that ξ and ξ1 are
bounded, the Cauchy–Schwartz inequality, the fact that the derivative of h0 is also
bounded, and since there exists a C > 0 such that Dφ1 (ξ2 ; ξ2 − ξ1 )2 ≤ C ξ2 − ξ1 X
for all ξ1 , ξ2 ∈ Xr . The last result follows from Theorem
3.3 and the boundedness of
thestate transition matrix. By an identical argument, Dψj,t (ξ1 ; ξ −ξ1 )−Dψj,t (ξ2 ; ξ −
ξ2 ) ≤ L ξ1 − ξ2 X . Finally, the result follows since Ψ+ (ξ) and Ψ− (ξ) are Lipschitz
continuous.
The following theorem shows that g is a well-defined function.
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Theorem 5.5. For each ξ ∈ Xp , the map ξ → ζ(ξ, ξ ) + V(ξ − ξ), with domain
Xr , is strictly convex and has a unique minimizer.
Proof. The map is strictly convex since it is the sum of the maximum of linear
functions of ξ − ξ (which is convex), the total variation of ξ − ξ (which is convex),
and the X -norm of ξ − ξ (which is strictly convex).
Next, let θ̄(ξ) = ζ(ξ, ξ ) + V(ξ − ξ) | ξ ∈ L2 ([0, 1], U ) × L2 ([0, 1], Σr ) . Note
that |ζ(ξ, ξ )| → ∞ as ξ L2 → ∞. Then, by Proposition II.1.2 in [10], there exists a
unique minimizer ξ ∗ of θ̄(ξ). But ξ ∗ ∈ Xr , since θ̄(ξ) and ζ(ξ, ξ ∗ ) are bounded. The
result follows after noting that Xr ⊂ L2 ([0, 1], U ) × L2 ([0, 1], Σr ); thus ξ ∗ is also the
unique minimizer of θ(ξ).
Finally, we prove that θ encodes a necessary condition for optimality.
Theorem 5.6. The function θ is nonpositive valued. Moreover, if ξ ∈ Xp is a
local minimizer of Pp as in Definition 4.4, then θ(ξ) = 0.
Proof. Notice that ζ(ξ, ξ) = 0 and θ(ξ) ≤ ζ(ξ, ξ), which proves the first part.
To prove the second part, we begin by making several observations. Given ξ ∈ Xr
and λ ∈ [0, 1], using the mean value theorem and condition (2) in Lemma 5.4 we have
that there exists L > 0 such that
(5.14)
2
J ξ + λ(ξ − ξ) − J(ξ) ≤ λ DJ ξ; ξ − ξ + Lλ2 ξ − ξX .
Similarly, if A(ξ) = arg max hj φt (ξ) | (j, t) ∈ J × [0, 1] , then there exists a pair
(j, t) ∈ A ξ + λ(ξ − ξ) such that using condition (3) in Lemma 5.4,
(5.15)
Ψ ξ + λ(ξ − ξ) − Ψ(ξ) ≤ λ Dψj,t ξ; ξ − ξ + Lλ2 ξ − ξ2X .
Also, by condition (1) in Assumption 3.4,
(5.16)
Ψ ξ + λ(ξ − ξ) − Ψ(ξ) ≤ L max φt ξ + λ(ξ − ξ) − φt (ξ)2 .
t∈[0,1]
We proceed by contradiction, i.e.,
we assume that θ(ξ) < 0 and show that for each
¯ ≤ 0 such that J(ξ)
ˆ < J(ξ). Note that
ε > 0 there exists ξˆ ∈ Nw (ξ, ε)∩ ξ̄ ∈ Xp | Ψ(ξ)
since θ(ξ) < 0, g(ξ) = ξ, and also, as a result of Theorem
4.2, for each ξ + λ(g(ξ)
−
ξ) ∈ Xr and ε > 0 there exists a ξλ ∈ Xp such that x(ξλ ) − x(ξ+λ(g(ξ)−ξ)) L2 <
ε . Now, letting ε = − λθ(ξ)
> 0, using Lemma 5.1, and adding and subtracting
2L
x(ξ+λ(g(ξ)−ξ)) ,
(5.17)
(ξ )
x λ − x(ξ) 2 ≤
L
θ(ξ)
+ L g(ξ) − ξX λ.
−
2L
Also, by (5.14), after adding and subtracting J ξ + λ(g(ξ) − ξ) ,
(5.18)
J(ξλ ) − J(ξ) ≤
θ(ξ)λ
+ 4A2 Lλ2 ,
2
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PART 1: A CONCEPTUAL ALGORITHM
2
where A = max u2 + 1 | u ∈ U and we used the fact that ξ − ξ X ≤ 4A2 and
DJ(ξ; ξ − ξ) ≤ θ(ξ). Hence for each λ ∈ 0, −θ(ξ)
8A2L , J(ξλ ) − J(ξ)
< 0. Using a similar
argument, and after adding and subtracting Ψ ξ + λ(g(ξ) − ξ) , we have
(5.19)
Ψ(ξλ ) ≤ L max φt (ξλ ) − φt ξ + λ(g(ξ) − ξ) 2 + Ψ(ξ) + θ(ξ) − γΨ(ξ) λ + 4A2 Lλ2
t∈[0,1]
≤
θ(ξ)λ
+ 4A2 Lλ2 + (1 − γλ)Ψ(ξ),
2
where we used the fact that Dψj,t (ξ; ξ − ξ) ≤ θ(ξ) − γΨ(ξ) for each j and t. Hence
1
for each λ ∈ 0, min −θ(ξ)
, Ψ(ξλ ) ≤ (1 − γλ)Ψ(ξ) ≤ 0.
8A2 L , γ
Summarizing, suppose ξ ∈ Xp is a local minimizer of Pp and θ(ξ) < 0. For each
ε > 0, by choosing any
−θ(ξ) 1
2Lε
,
,
(5.20)
λ ∈ 0, min
,
8A2 L γ 2L2 g(ξ) − ξX − θ(ξ)
we can construct a ξλ ∈ Xp such that ξλ ∈ Nw (ξ, ε), J(ξλ ) < J(ξ), and Ψ(ξλ ) ≤ 0.
Therefore, ξ is not a local minimizer of Pp , which is a contradiction.
5.2. Approximating relaxed inputs. In this subsection, we prove that ρN
takes elements from the relaxed optimization space to the pure optimization space
with a known bound for the quality of approximation.
Lemma 5.7. For each N ∈ N and d ∈ Dr , FN (d) ∈ Dr and PN FN (d) ∈ Dp .
Proof. Note first that [FN (d)]i (t) ∈[0, 1] for each t ∈ [0, 1] due to the result in
q
section 3.3 in [14]. Next, observe that i=1 [FN (d)]i (t) = 1 for each t ∈ [0, 1] since
the wavelet approximation is linear and
(5.21)
q
i=1
[FN (d)]i =
q
k
N 2
−1
di , bk,j di , 1 +
i=1
k=0 j=0
bk,j
2
bk,j L2
k
N 2
−1
1, bk,j = 1, 1 +
k=0 j=0
bk,j
2
bk,j L2
= 1,
where the last equality holds since 1, bk,j = 0 for each k, j.
Next, observe that PN FN (d) i (t) ∈ {0, 1} for each t ∈ [0, 1] due to the defini
q tion of PN . Also note that i=1 PN FN (d) i (t) = 1 for each t ∈ [0, 1]. This follows
because
of the piecewise continuity of FN (d) over intervals of size 2−N and because
q
i=1 [FN (d)]i (t) = 1, which implies that {[SN,k,i−1 , SN,k,i )}(i,k)∈{1,...,q}×{0,...,2N −1}
forms a partition of [0, 1]. Our desired result follows.
The next two lemmas lay the foundations of our extension to the chattering
lemma and rely upon a bounded variation assumption in order to construct a rate of
approximation, which is critical while choosing a value for the parameter β̄ in order to
ensure that if θ(ξ) < 0, then ν(ξ) < ∞. This is proved in Lemma 5.14 and guarantees
that step 7 of Algorithm 4.1 is well-defined.
Lemma 5.8. Let f ∈ L2 ([0, 1], R) ∩ BV ([0, 1], R); then
(5.22)
f − FN (f )L2
1
≤
2
1
√
2
N
V(f ).
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Proof. Since L2 is a Hilbert space and the collection {bk,j }k,j is a basis, then
k
f = f, 1 +
(5.23)
∞ 2
−1
f, bk,j k=0 j=0
bk,j
2
bk,j L2
.
t
2
Note that bk,j L2 = 2−k and that 0bk,j (s)ds ≤ 2−k−1
for each t ∈ [0, 1]. Thus, by
Proposition 3.1, f, bk,j ≤ 2−k−1 V f 1[j2−k ,(j+1)2−k ] . Finally, Parseval’s identity
for Hilbert spaces (Theorem 5.27 in [11]) implies that
(5.24) f −
2
FN (f )L2
≤
∞
k=N +1
2
−k−2
k
−1
2
2
2
V f 1[j2−k ,(j+1)2−k ]
≤ 2−N −2 V(f ) ,
j=0
as desired, where the last inequality follows by the definition in (3.3).
Lemma 5.9. There exists K > 0 such that for each d ∈ Dr and f ∈ L2 ([0, 1], Rq )∩
BV ([0, 1], Rq ),
N
N
1
1
√
d − PN FN (d) , f ≤ K
f L2 V(d) +
V(f ) .
(5.25)
2
2
i
Proof. To simplify our notation, let pi,k = [FN (d)]i 2kN , Si,k = j=1 pj,k , and
t
p
k+Si−1,k k+Si,k Ai,k =
, 2N . Note that k pi,k − 1Ai,k (s)ds ≤ 2i,k
N for each t ∈ [0, 1].
2N
2N
Thus, by Proposition 3.1, the definition in (3.3), and Hölder’s inequality,
(5.26)
N −1 q 2
FN (d) − PN FN (d) , f = k=0 i=1
k+1
2N
k
2N
1
pi,k − 1Ai,k (t) fi (t)dt ≤ N V(f ).
2
The final result follows from the inequality above, Lemma 5.8, and the Cauchy–
Schwartz inequality.
The following theorem is an extension of the chattering lemma for functions of
bounded variation. As explained earlier, this theorem is key to our result since it
provides a rate of convergence for the approximation of trajectories.
Theorem 5.10. There exists K > 0 such that for each ξ ∈ Xr and t ∈ [0, 1],
(5.27)
φt ρN (ξ) − φt (ξ) ≤ K
2
1
√
2
N
V(ξ) + 1 .
Proof. Let ξ = (u,
d). To simplify our notation, let us denote uN = FN (u)
and dN = PN FN (d) ; thus ρN (ξ) = (uN , dN ). First note that since f is Lipschitz
continuous,
(5.28)
1
(u ,d )
(u ,d )
x N N (t) − x(u,dN ) (t) ≤ L
x N N (s) − x(u,dN ) (s) + uN (s) − u(s) ds
2
2
2
0
√
N
1
LeL 2
√
V(u),
≤
2
2
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4479
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where the last inequality follows by Bellman–Gronwall’s inequality together with
Lemma 5.8. On the other hand,
(5.29)
(u,d )
x N (t) − x(u,d) (t) ≤ eL 2
1
0
q
(u,d)
[dN ]i (s) − di (s) f t, x
(s), u(s), ei ds
,
2
i=1
where again we
continuity of f and Bellman–Gronwall’s inequality.
used the Lipschitz Let vk,i (t) = f t, x(u,d) (t), u(t), ei k and vk = (vk,1 , . . . , vk,q ); then vk is of bounded
variation. Indeed, x(ξ) is of bounded variation
Corollary 3.33 and Lemma 3.34
(by
in [11]); thus there exists C > 0 such that V x(ξ) ≤ C and vk,i L2 ≤ C, and using
the Lipschitz continuity of f , we get that for each i ∈ Q, V(vk,i ) ≤ L 1 + C + V(u) .
Hence, Lemma 5.9 implies that there exists K > 0 such that for each k ∈ {1, . . . , n},
N
N
1
1
√
d − dN , vk ≤ K
(5.30)
1 + C + V(u) .
CV(d) +
2
2
The result follows easily from these three inequalities.
5.3. Convergence of the algorithm. To prove the convergence of our algorithm, we employ a technique similar to the one prescribed in section 1.2 in [22].
Summarizing the technique, one can think of an algorithm as a discrete-time dynamical system, whose desired stable equilibria are characterized by the stationary points
of its optimality function, i.e., points ξ ∈ Xp , where θ(ξ) = 0, since we know from
Theorem 5.6 that all local minimizers are stationary. Before applying this line of reasoning to our algorithm, we present a simplified version of this argument for a general
unconstrained optimization problem in the interest of clarity.
Definition 5.11 (Definition 2.1 in [3]2 ). Let S be a metric space, and consider
the problem of minimizing the cost function J : S → R. A function Γ : S → S has
the uniform sufficient descent property with respect to an optimality function θ : S →
(−∞, 0] if for each C > 0 there exists a δC > 0 such that for every x ∈ S with
θ(x) < −C, J Γ(x) − J(x) ≤ −δC .
A sequence of points generated by an algorithm satisfying the uniform sufficient
descent property can be shown to approach the zeros of the optimality function.
Theorem 5.12 (Proposition 2.1 in [3]). Let S be a metric space and J : S → R
be a lower bounded function. Suppose that Γ : S → S satisfies the uniform sufficient
descent property with respect to θ : S → (−∞, 0]. If {xj }j∈N is constructed such that
(5.31)
xj+1 =
Γ(xj )
xj
if θ(xj ) < 0,
if θ(xj ) = 0,
then limj→∞ θ(xj ) = 0.
Proof. Suppose that lim inf j→∞ θ(xj ) = −2ε < 0. Then there exists a subsequence {xjk }k∈N such that θ(xjk ) < −ε for each k ∈ N. Definition 5.11 implies that
there exists δε such that for each k ∈ N, J(xjk +1 ) − J(xjk ) ≤ −δε . But this is a contradiction, since J(xj+1 ) ≤ J(xj ) for each j ∈ N, and thus J(xj ) → −∞ as j → ∞,
even though J is lower bounded.
2 Versions
of this result can also be found in [23, 35].
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4480
VASUDEVAN, GONZALEZ, BAJCSY, AND SASTRY
Note that Theorem 5.12 does not assume the existence of accumulation points of
the sequence {xj }j∈N . This is critical in infinite-dimensional optimization problems
where the level sets of a cost function may not be compact. Our proof of convergence
of the sequence of points generated by Algorithm 4.1 does not make explicit use of
Theorem 5.12 since ρN and the constraints require special treatment, but the main
argument of the proof follows from a similar principle.
We begin by showing that μ and ν are bounded when θ is negative.
Lemma 5.13. Let α, β ∈ (0, 1). For each δ > 0 there exists Mδ∗ < ∞ such that if
θ(ξ) ≤ −δ for ξ ∈ Xp , then μ(ξ) ≤ Mδ∗ .
Proof. Assume that Ψ(ξ) ≤ 0. Recall that DJ ξ; g(ξ) − ξ ≤ θ(ξ); then using
(5.14),
(5.32)
J ξ + β k (g(ξ) − ξ) − J(ξ) − αβ k θ(ξ) ≤ −(1 − α)β k θ(ξ) + 4A2 Lβ 2k ,
where A = max
u
+
1
|
u
∈
U
. Hence, for each k ∈ N such that βk ≤ (1−α)δ
2
4A2 L we
k
k
have that J ξ + β (g(ξ) − ξ) − J(ξ) ≤ αβ θ(ξ). Similarly, since Dψj,t ξ; g(ξ) − ξ ≤
θ(ξ) for each (j, t) ∈ J × [0, 1], using (5.15),
(5.33)
Ψ ξ + β k (g(ξ) − ξ) − Ψ(ξ) + β k γΨ(ξ) − αθ(ξ) ≤ −(1 − α)β k θ(ξ) + 4A2 Lβ 2k ;
hence, for each k ∈ N such that β k ≤ min
1 − β k γ Ψ(ξ) ≤ 0.
Now, if Ψ(ξ) > 0,
(5.34)
(1−α)δ
4A2 L
, γ1 , Ψ ξ+β k (g(ξ)−ξ) −αβ k θ(ξ) ≤
Ψ ξ + β k (g(ξ) − ξ) − Ψ(ξ) − αβ k θ(ξ) ≤ −(1 − α)δβ k + 4A2 Lβ 2k .
k
k
Hence, for each k ∈ N such that β k ≤ (1−α)δ
4A2 L , Ψ ξ + β (g(ξ) − ξ) − Ψ(ξ) ≤ αβ θ(ξ).
1
, then we get that μ(ξ) ≤ Mδ∗
Finally, let Mδ∗ = 1 + max logβ (1−α)δ
4A2 L , logβ γ
as desired.
The following result relies on the rate of convergence result proved in Theorem 5.10
that follows from Lemmas 5.8 and 5.9 and from the assumption that the relaxed
optimization space is of bounded variation.
Lemma 5.14. Let α, β, ω ∈ (0, 1), ᾱ ∈ (0, ∞), β̄ ∈ √12 , 1 , and ξ ∈ Xp . If
θ(ξ) < 0, then ν(ξ) < ∞.
Proof. To simplify our notation let us denote M = μ(ξ) and ξ = ξ +β M g(ξ)−ξ .
Theorem 5.10 implies that there exists K > 0 such that
(5.35)
N J ρN (ξ ) − J(ξ ) ≤ K L 2− 2 V(ξ ) + 1 ,
where L is the Lipschitzconstant
of h0 .
Let A(ξ) = arg max hj φt (ξ) | (j, t) ∈ J × [0, 1] ; then given (j, t) ∈ A ρN (ξ ) ,
(5.36)
N Ψ ρN (ξ ) − Ψ(ξ ) ≤ ψj,t ρN (ξ ) − ψj,t (ξ ) ≤ K L 2− 2 V(ξ ) + 1 .
N Hence, there exists N0 ∈ N such that K L 2− 2 V(ξ ) + 1 ≤ −ᾱβ̄ N θ(ξ) for each
N ≥ N0 . Also, there exists N1 ≥ N0 such that for each N ≥ N1 , ᾱβ̄ N ≤ (1 − ω)αβ M .
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Now suppose that Ψ(ξ) ≤ 0; then for each N ≥ N1 the following inequalities hold:
(5.37) J ρN (ξ ) − J(ξ) = J ρN (ξ ) − J(ξ ) + J(ξ ) − J(ξ) ≤ αβ M − ᾱβ̄ N θ(ξ),
(5.38)
Ψ ρN (ξ ) = Ψ ρN (ξ ) − Ψ(ξ ) + Ψ(ξ ) ≤ αβ M − ᾱβ̄ N θ(ξ) ≤ 0.
Similarly, if Ψ(ξ) > 0, then, using the same argument as above, we have that
(5.39)
Ψ ρN (ξ ) − Ψ(ξ) ≤ αβ M − ᾱβ̄ N θ(ξ).
Therefore, from (5.37), (5.38), and (5.39), it follows that ν(ξ) ≤ N1 as desired.
The following lemma proves that once Algorithm 4.1 finds a feasible point, every
point generated afterward is also feasible. We omit the proof since it follows directly
from the definition of ν.
Lemma 5.15. Let {ξi }i∈N be a sequence generated by Algorithm 4.1. If there
exists i0 ∈ N such that Ψ(ξi0 ) ≤ 0, then Ψ(ξi ) ≤ 0 for each i ≥ i0 .
Employing these preceding results, we can prove that every sequence produced
by Algorithm 4.1 asymptotically satisfies our optimality condition.
Theorem 5.16. Let {ξi }i∈N be a sequence generated by Algorithm 4.1. Then
limi→∞ θ(ξi ) = 0.
Proof. If the sequence produced by Algorithm 4.1 is finite, then the theorem is
trivially satisfied, so we assume that the sequence is infinite. Suppose the theorem is
not true; then lim inf i→∞ θ(ξi ) = −2δ < 0 and therefore there exists k0 ∈ N and a
subsequence {ξik }k∈N such that θ(ξik ) ≤ −δ for each k ≥ k0 . Also, recall that ν(ξ)
was chosen such that given μ(ξ), αβ μ(ξ) − ᾱβ̄ ν(ξ) ≥ ωαβ μ(ξ) .
From Lemma 5.13 we know that there exists Mδ∗ , which depends on δ, such that
∗
μ(ξ)
β
≥ β Mδ . Suppose that the subsequence {ξik }k∈N is eventually feasible; then by
Lemma 5.15, the sequence is always feasible. Thus,
(5.40)
∗
J Γ(ξik ) − J(ξik ) ≤ αβ μ(ξ) − ᾱβ̄ ν(ξ) θ(ξik ) ≤ −ωαβ μ(ξ) δ ≤ −ωαβ Mδ δ.
This inequality, together with the fact that J(ξi+1 ) ≤ J(ξi ) for each i ∈ N, implies
that lim inf k→∞ J(ξik ) = −∞, but this is a contradiction since J is lower bounded,
which follows from Theorem 3.3.
The case when the sequence is never feasible is analogous after noting that since
the subsequence is infeasible, then Ψ(ξik ) > 0 for each k ∈ N, establishing a similar
contradiction.
As described in the discussion that follows Theorem 5.12, this result does not
prove the existence of accumulation points to the sequence generated by Algorithm 4.1,
but only shows that the generated sequence asymptotically satisfies a necessary condition for optimality.
Acknowledgments. We are grateful to Maryam Kamgarpour and Claire Tomlin, who helped write our original papers on switched system optimal control. We are
also grateful to Magnus Egerstedt and Yorai Wardi, whose comments on our original
papers were critical to our realizing the deficiencies of the sufficient descent argument.
The chattering lemma argument owes its origin to a series of discussions with Ray
DeCarlo. The contents of this article reflect a longstanding collaboration with Elijah
Polak.
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