Mem. Differential Equations Math. Phys. 29(2003), 153–155
L. Alkhazishvili
THE LINEARIZED MAXIMUM PRINCIPLE FOR OPTIMAL
PROBLEMS WITH VARIABLE DELAYS AND CONTINUOUS INITIAL
CONDITION
(Reported on March 3, 2003)
Optimal problems with variable delays in phase coordinates and controls are considered. Without commensurability conditions for delays in controls (incommensurability),
necessary conditions of optimality are obtained: in the form of the linearized integral
maximum principle for initial function and control, in the form of equalities and inequalities for initial and final moments.
Let J = [a, b] be a finite interval; O ⊂ Rn , G ⊂ Rn be open sets and let the function
f : J × O s × Gν → Rn satisfies the following conditions:
1. for a fixed t ∈ J , the function f (t, x1 , . . . , xs , u1 , . . . , uν ) is continuously differentiable with respect to (x1 , . . . , xs , u1 , . . . , uν ) ⊂ O s × Gν ;
2. for a fixed (x1 , . . . , xs , u1 , . . . , uν ) ⊂ O s × Gν , the functions f , fxi , i = 1, . . . , s,
fuj , j = 1, . . . , ν, are measurable with respect to t. For arbitrary compacts
+
K ⊂ O, V ⊂ G, there exists a function mK,V (·) ∈ L(J , R+
0 ), R0 = [0, ∞),
s
such that for almost all t ∈ J and ∀(x1 , . . . , xs , u1 , . . . , uν ) ∈ K × V ν ,
|f (t, x1 , . . . , xs , u1 , . . . , uν )| +
s
X
i=1
|fxi (·)| +
ν
X
|fuj (·)| ≤ mK,V (t).
j=1
Let now τi (t), i = 1, . . . , s, θj (t), j = 1, . . . , ν, t ∈ J , are absolutely continuous
functions satisfying the conditions: τi (t) ≤ t, τ̇i (t) > 0, θj (t) ≤ t, θ̇j (t) ≥ 0; ∆ be a set
of continuous functions ϕ : [τ, b] → M , τ = min{τ1 (a), . . . , τs (a)}, M ⊂ O is a convex
set; Ω be a set of measurable functions u : J2 = [θ, b] → U , θ = min{θ1 (a), . . . , θν (a)}
satisfying the conditions cl{u(t), t ∈ J2 } is compact lying in G, U ⊂ G is a convex set;
q i (t0 , t1 , x0 , x1 ), i = 0, . . . , l, (t0 , t1 , x0 , x1 ) ∈ J 2 × O 2 , are continuously differentiable
scalar functions.
We consider the differential equation in Rn
ẋ(t) = f (t, x(τ1 (t)), . . . , x(τs (t)), u(θ1 (t)), . . . , u(θν (t))),
(1)
with the continuous condition
x(t) = ϕ(t),
t ∈ [τ, t0 ].
(2)
Definition 1. The function x(t) = x(t, σ) ⊂ O, σ = (t0 , t1 , ϕ(·), u(·)) ∈ A =
J 2 × ∆ × Ω, t0 < t1 , defined on the interval [τ, t1 ], is said to be a solution corresponding
to the element σ ∈ A if on the interval [τ, t0 ] it satisfies the condition (2), while on
the interval [t0 , t1 ] the trajectory x(t) is absolutely continuous and almost everywhere
satisfies the equation (1).
2000 Mathematics Subject Classification. 49K25.
Key words and phrases. Optimal control problem, necessary condition of optimality,
delay.
154
Definition 2. The element σ ∈ A is said to be admissible if the corresponding solution
x(t) satisfies the conditions
q i (t0 , t1 , x(t0 ), x(t1 )) = 0,
i = 1, . . . , l.
The set of admissible elements will be denoted by A0 .
Definition 3. The element σ
e = (e
t0 , e
t1 , ϕ(·),
e
u
e(·)) ∈ A0 is said to be optimal if for an
arbitrary element σ ∈ A0 the inequality
holds.
q 0 (e
t0 , e
t1 , x
e(e
t0 ), x
e(e
t1 )) ≤ q 0 (t0 , t1 , x(t0 ), x(t1 )),
x
e(t) = x(t, σ
e),
The problem of optimal control consists in finding an optimal element.
In order to formulate the main results, consider the following notation:
ω = (t, x1 , . . . , xs ) ∈ J × O s ,
ω0 = (e
t0 , x
e(τ1 (e
t0 )), . . . , x
e(τs (e
t0 ))),
γi (t) = τi−1 (t),
ω1 = (e
t1 , x
e(τ1 (e
t1 )), . . . , x
e(τs (e
t1 ))),
fe[ω] = f (t, x1 , . . . , xs , u
e(θ1 (t)), . . . , u
e(θν (t))),
fexi [t] = fxi (t, x
e(τ1 (t)), . . . , x
e(τs (t)), u
e(θ1 (t)), . . . , u
e(θν (t))),
feuj [t] = fuj (t, x
e(τ1 (t)), . . . , x
e(τs (t)), u
e(θ1 (t)), . . . , u
e(θν (t))),
R−
t = (−∞, t].
Theorem 1. Let σ
e ∈ A0 be an optimal element, e
t0 ∈ (a, b), e
t1 ∈ (a, b] and the
following conditions hold:
a) the function ϕ(t)
e
is absolutely continuous in some left semi-neighborhood of the
point e
t0 ;
b) there exist the finite limits:
ϕ̇− = ϕ(
ė e
t−
0 );
lim fe[ω] = f0− ,
ω→ω0
lim fe[ω] = f1− ,
ω ∈ R−
× Os ;
e
t0
ω→ω1
ω ∈ R−
× Os .
e
t1
Then there exists a non-zero vector π = (π0 , . . . , πl ), π0 ≤ 0, and a solution ψ(t),
t ∈ [e
t0 , γ], γ = max(γ1 (b), . . . , γs (b)), of the equation
ψ̇(t) = −
s
X
i=1
ψ(γi (t))fexi [γi (t)]γ̇i (t),
t ∈ (e
t1 , γ],
ψ(t) = 0,
t ∈ [e
t0 , e
t1 ],
(3)
such that the following conditions are fulfilled
Zte0
s
X
i=1
τi (te0 )
≥
ψ(t)fexi [γi (t)]γ̇i (t)ϕ(t)
e dt ≥
Zte0
s
X
i=1
τi (te0 )
ν
X
ψ(t)fexi [γi (t)]γ̇i (t)ϕ(t) dt,
∀ϕ(·) ∈ ∆;
e x + ψ(e
e x + ψ(e
t0 ) ϕ(
ee
t0 ) ≥ π Q
t0 ) ϕ,
πQ
0
0
Zte1
j=1 e
t0
ψ(t)feuj [t]e
u(θj (t)) dt ≥
e t ≥ −ψ(e
πQ
t1 )f1− ;
1
ν
X
Zte1
j=1 e
t0
∀ϕ ∈ M ;
ψ(t)feuj [t]u(θj (t)) dt,
et + Q
e x ϕ̇− ) ≥ ψ(e
π(Q
t0 )(f0− − ϕ̇− ).
0
0
(4)
(5)
∀u(·) ∈ Ω;
(6)
155
Here the tilde over Q = (q 0 , . . . , q l )> means the the corresponding gradient is calculated
at the point (e
t0 , e
t1 , x
e(e
t0 ), x
e(e
t1 )).
Theorem 2. Let σ
e ∈ A0 be an optimal element, e
t0 ∈ [a, b), e
t1 ∈ (a, b) and the
following conditions hold:
c) the function ϕ(t)
e
is absolutely continuous in some right semi-neighborhood of the
point e
t0 ;
d) there exist the finite limits:
ϕ̇+ = ϕ(
ė e
t+
0 );
lim fe[ω] = f0+ ,
ω→ω0
ω ∈ R+
× Os ;
e
t0
lim fe[ω] = f1+ ,
ω→ω1
ω ∈ R+
× Os .
e
t1
Then there exists a non-zero vector π = (π0 , . . . , πl ), π0 ≤ 0, and a solution ψ(t) of the
equation (3) such that (4)–(6) are fulfilled. Moreover,
e t ≤ −ψ(e
πQ
t1 )f1+ ;
1
et + Q
e x ϕ̇+ ) ≤ ψ(e
π(Q
t0 )(f0+ − ϕ̇+ ).
0
0
Theorem 3. Let σ
e ∈ A0 be an optimal element, e
t0 , e
t1 ∈ (a, b) and the assumptions
of Theorems 1, 2 hold. Let, besides,
e) ϕ̇− = ϕ̇+ = ϕ̇, f0− = f0+ = f0 ;
f) f1− = f1+ = f1 .
Then there exists a non-zero vector π = (π0 , . . . , πl ), π0 ≤ 0, and a solution ψ(t) of the
equation (3) such that (4)–(6) are fulfilled. Moreover,
e t = −ψ(e
πQ
t1 )f1 ;
1
et + Q
e x ϕ̇) = ψ(e
π(Q
t0 )(f0 − ϕ̇).
0
0
e x ) = 1 + l, then in Theorem 3 ψ(t) 6≡ 0.
et , Q
ex , Q
et , Q
We would note that if rank(Q
1
0
1
0
Optimal problems of various classes with commensurable and incommensurable delays
in control are considered in [1]–[6].
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Equations Math. Phys. 22(2001), 154–158.
Author’s address:
Department of Applied Mathematics and Computer Sciences
I. Javakhishvili Tbilisi State University
2, University St., Tbilisi 380043, Georgia