Mem. Differential Equations Math. Phys. 31(2004), 145–148
I. Ramishvili
THE LINEARIZED MAXIMUM PRINCIPLE FOR QUASI-LINEAR
NEUTRAL OPTIMAL PROBLEMS WITH DISCONTINUOUS INITIAL
CONDITION AND VARIABLE DELAYS IN CONTROLS
(Reported on November 17, 2003)
Let J = [a, b] ⊂ R be a finite interval, O ⊂ Rn , G ⊂ Rr be open sets; Let the
function f : J × O s × Gν → Rn satisfy the following conditions: for almost all t ∈ J
function f (t, x1 , . . . , xs , u1 , . . . , uν ) is continuously differentiable with respect to xi ∈
O, i = 1, . . . , s, um ∈ G, m = 1, . . . , ν; for any fixed (x1 , . . . , xs , u1 , . . . , uν ) ∈ O s ×
Gν the functions f (t, x1 , . . . , xs , u1 , . . . , uν ), fxi (·) i = 1, . . . , s, fum (·), m = 1, . . . , ν
are measurable on J; for arbitrary compacts K ⊂ O, N ⊂ G there exists a function
mK,N (·) ∈ L(J, R+ ), R+ = [0, ∞), such that for any (x1 , . . . , xs , u1 , . . . , um ) ∈ K s ×N m
and for almost all t ∈ J, the following inequality is fulfilled
| f (t, x1 , . . . , xs , u1 , . . . , uν ) | +
s
X
i=1
| fxi (·) | +
ν
X
| fum (·) |≤ mK,N (t).
m=1
Let the scalar functions τi (t), i = 1, . . . , s, θm (t), m = 1, . . . , ν, t ∈ R and ηj (t),
j = 1, . . . , k, t ∈ R, be absolutely continuous and continuously differentiable , respectively,
and satisfy the conditions: τi (t) ≤ t, τ̇i (t) > 0, i = 1, . . . , s; θm (t) ≤ t, θ̇m (t) > 0,
m = 1, . . . , ν ηj (t) < t, η̇j (t) > 0, j = 1, . . . , k. Let Φ be the set of continuously
differentiable functions ϕ : J1 = [τ, b] → M, τ = min{η1 (a), . . . , ηk (a), τ1 (a), . . . , τs (a)},
where M ⊂ O is a convex set, k ϕ k= sup{| ϕ(t) | + | ϕ̇(t) |: t ∈ J}; Ω be the set of
measurable functions u : J2 = [θ, b] → U, such that cl{u(t) : t ∈ J} is a compact lying in
G, θ = min{θ1 (a), . . . , θν (a)}, where U ⊂ G is a convex set, k u k= sup{| u(t) |: t ∈ J2 };
Ai (t), t ∈ J, i = 1, . . . , k, be continuous n × n matrix functions. The scalar functions
q i (t0 , t1 , x0 , x1 ), i = 1, . . . , l, are continuously differentiable on the set J 2 × O 2 .
To every element λ = (t0 , t1 , x0 , ϕ, u) ∈ E = J 2 × O × Φ × Ω let us correspond the
differential equation
ẋ(t) =
k
X
Aj (t)ẋ(ηj (t)) + f (t, x(τ1 (t)), . . . , x(τs (t)), u(θ1 (t)), . . . , u(θm (t)))
(1)
j=1
with discontinuous initial condition
x(t) = ϕ(t), t ∈ [τ, t0 ), x(t0 ) = x0 .
(2)
Definition 1. Let λ = (t0 , t1 , x0 , ϕ, u) ∈ E, t0 < b. The function x(t) = x(t; λ) ∈ O,
t ∈ [τ, t1 ], t1 ∈ (t0 , b] is said to be a solution corresponding to the element λ, defined on
the interval [τ, t1 ], if on the interval [τ, t0 ] the function x(t) satisfies the condition (2),
while on the interval [t0 , t1 ] it is absolutely continuous and almost everywhere satisfies
the equation (1).
2000 Mathematics Subject Classification. 49K25.
Key words and phrases. Neutral differential equation, Necessary conditions of
optimality.
146
Definition 2. The element λ ∈ E is said to be admissible if the corresponding solution
x(t) = x(t; λ) satisfies the conditions
q i (t0 , t1 , x0 , x(t1 )) = 0,
i = 1, . . . , l.
(3)
The set of admissible elements will be denoted by E0 .
e = (e
Definition 3. The element λ
t0 , e
t1 , x
e0 , ϕ,
e u
e) ∈ E0 is said to be locally optimal, if
there exists a number δ > 0 such that for an arbitrary element λ ∈ E0 satisfying
|e
t0 − t0 | + | e
t1 − t1 | + | x
e0 − x0 | + k ϕ
e−ϕ k + k u
e − u k≤ δ
the inequality
q 0 (e
t0 , e
t1 , x
e0 , x
e(e
t1 )) ≤ q 0 (t0 , t1 , x0 , x(t1 ))
(4)
e
holds, where x
e(t) = x(t; λ).
The problem (1)-(4) is said to be optimal problem with discontinuous initial condition
and it consists in finding a locally optimal element.
In order to formulate the main results, we will introduce the following notation
σi = (te0 , x
f0 , . . . , x
f0 , ϕ(
e te0 ), . . . , ϕ(
e te0 ), ϕ(τ
e p+1 (te0 )), . . . , ϕ(τ
e s (te0 ))), i = 0, . . . , p ;
| {z } |
{z
}
i
(p−i)
σi = (γi , x
e(τ1 (γi )), . . . , x
e(τi−1 (γi )), x
e0 , ϕ(τ
e i+1 (γi )), . . . , ϕ(τ
e s (γi ))),
σi0 = (γi , x
e(τ1 (γi )), . . . , x
e(τi−1 (γi )), ϕ(
ee
t0 ), ϕ(τ
e i+1 (γi )), . . . , ϕ(τ
e s (γi ))),
i = p + 1, . . . , s; σs+1 = (e
t1 , x
e(τ1 (e
t1 )), . . . , x
e(τs (e
t1 ))), γi = γi (e
t0 ),
ρj = ρj (te0 ), γi (t) = τi−1 (t), ρj (t) = ηj−1 (t); ω = (t, x1 , . . . , xs ),
fe(ω) = f (ω, u
e(θ1 (t)), . . . , u
e(θν (t))),
e(τ1 (t)), . . . , x
e(τs (t)), . . . , u
e(θ1 (t)), . . . , u
e(θν (t)));
ff
xi [t] = fxi (t, x
Q = (q 0 , . . . , q l )0 ,
e t = Qt (te0 , te1 , x
Q
e0 , x
e(e
t1 )).
0
0
e = (e
Theorem 1. Let the element λ
t0 , te1 , x
e0 , ϕ,
e u
e) ∈ E0 , e
t0 > a be locally optimal
and the following conditions be fulfilled:
1) γi = e
t0 , i = 1, . . . , p γp+1 < · · · < γs < e
t 1 , ρj < e
t1 j = 1, . . . , k;
2) there exists a number δ > 0 such that
t ∈ (te0 − δ, te0 ];
γ1 (t) ≤ · · · ≤ γp (t),
3) there exist the finite limits: γ̇i− = γ̇i (e
t0 −), i = 1, . . . , s, x(η
ė j (e
t1 −)), j = 1, . . . , k;
lim fe(ω) = fi− , ω ∈ (e
t0 − δ, e
t0 ] × O s , i = 0, . . . , p,
i
fe(ω1 ) − fe(ω2 ) = fi− , ω1 , ω2 ∈ (γi − δ, γi ] × O s , i = p + 1, . . . , s,
ω→σi
lim
(ω1 ,ω2 )→(σi ,σi0 )
h
lim
ω→σs+1
−
fe(ω) = fs+1
,
ω ∈ (e
t1 − δ, e
t1 ] × O s .
Then there exist a non-zero vector π = (π0 , . . . , πl ), π0 ≤ 0, and a solution χ(t) =
(χ1 (t), . . . , χn (t)), ψ(t) = (ψ1 (t), . . . , ψn (t)) of the system

Ps
e

χ̇(t) = − i=1Pψ(γi (t))fxi [γi (t)]γ̇i (t),
(5)
ψ(t) = χ(t) + kj=1 ψ(ρj (t)Aj (ρj (t))ρ̇j (t), t ∈ [e
t0 , e
t1 ],


ψ(t) = 0, t > e
t1 ,
such that the following conditions are fulfilled:
147
a) the linearized maximum principles:
Z te1
Z
ν
X
ψ(t)
feum [t]e
u(θm (t))dt ≥
te0
Z
s
X
i=p+1
m=1
e
t0
τi (te0 )
te1
e
t0
ψ(γi (t))fexi [γi (t)]γ̇i (t)ϕ(t)dt
e
+
Z
s
X
≥
i=p+1
+
k Z
X
j=1
te0
ηj (te0 )
te0
τi (te0 )
ψ(t)
ν
X
m=1
k Z
X
j=1
e
t0
feum [t]u(θm (t))dt,
ηj (te0 )
ψ(ρj (t))Aj [ρj (t)]ρ̇j (t)ϕ(t)dt
ė
≥
ψ(γi (t))fexi [γi (t)]γ̇i (t)ϕ(t)dt+
ψ(ρj (t))Aj [ρj (t)]ρ̇j (t)ϕ̇(t)dt,
b) the conditions for the moments e
t0 , e
t1 :
e t ≥ −ψ(e
πQ
t0 −)[ϕ(
ė e
t0 ) −
0
+
s
X
k
X
j=1
Aj (e
t0 )ϕ(η
ė j (e
t0 )) +
ψ(γi −)fi− γ̇i−
p+1
e t ≥ −ψ(e
πQ
t1 )
1
∀u ∈ Ω,
X
k
j=1
∀ϕ ∈ Φ;
p
X
−
(b
γi+1
−γ
bi− )fi− ]+
i=0
+ χ(e
t0 )ϕ(
ė e
t0 ),
−
Aj (e
t1 )x(η
ė j (e
t1 −)) + fs+1
;
c) the condition for the solution χ(t), ψ(t)
e x = −χ(e
t0 ),
πQ
0
Here
γ
b0− = 1, γ
bi− = γ̇i− ,
e x = ψ(e
t1 ) = χ(e
t1 ).
πQ
1
i = 1, . . . , p,
−
γ
bp+1
= 0;
e = (e
Theorem 2. Let the element λ
t0 , e
t1 , x
e0 , ϕ,
e u
e) ∈ E0 , e
t1 < b be locally optimal and
the condition 1) of Theorem 1 and the following conditions be fulfilled:
4) there exists a number δ > 0 such that
γ1 (t) ≤ · ≤ γp (t),
t ∈ [te0 , te0 + δ);
5) there exist the finite limits : γ̇i+ = γ̇i (e
t0 +), i = 1, . . . , s, x(η
ė j (e
t1 +)), j = 1, . . . , k;
lim fe(ω) = fi+ , ω ∈ [e
t0 , e
t0 + δ) × O s , i = 0, . . . , p,
h
i
fe(ω1 ) − fe(ω2 ) = fi+ , ω1 , ω2 ∈ [γi , γi + δ) × O s , i = p + 1, . . . , s
ω→σi
lim
(ω1 ,ω2 )→(σi ,σi0 )
lim
ω→σs+1
+
fe ω) = fs+1
, ω ∈ [e
t1 , e
t1 + δ × O s .
Then there exists a non-zero vector π = (π0 , . . . , πl ), π0 ≤ 0 and a solution χ(t), ψ(t) of
the system (5) such that the conditions a) and c) are fulfilled. Moreover,
e t ≤ −ψ(e
t0 +)[ϕ(
ė e
t0 ) −
πQ
0
+
s
X
i=p+1
k
X
j=1
Aj (e
t0 )ϕ(η
ė j (e
t0 )) +
p
X
i=0
+
(b
γi+1
−γ
bi+ )fi+ ]+
ψ(γi +)fi+ γ̇i+ + χ(e
t0 )ϕ(
ė e
t0 ),

e t ≤ −ψ(e
πQ
t1 ) 
0
k
X
j=1
Aj (e
t1 )x(η
ė j (e
t1 +)) +

+ 
.
fs+1
148
Here
+
γ
b0+ = 1, γ
bi+ = γ̇i+ , i = 1, . . . , p, γ
bp+1
= 0.
e = (e
Theorem 3. Let the element λ
t0 , e
t1 , x
e0 , ϕ,
e u
e) ∈ E0 , e
t0 , e
t1 ∈ (a, b) be locally
optimal and the conditions of Theorems 1, 2 and the following conditions be fulfilled:
the functions x(η
ė j (e
t1 )), j = 1, . . . , k, are continuous;
γi , e
t0 ∈
/ ηk1 (ηk2 (. . . (ηke (e
t1 )), . . . , )) ∈ (a, e
t1 ) : e = 1, 2, . . . , m = 1, . . . , e,
km = 1, . . . , k , i = p + 1, . . . , s;
p
X
i=0
−
(b
γi+1
−γ
bi− )fi− =
fi− γ̇i− = fi+ γ̇i+ = fi ,
p
X
i=0
+
(b
γi+1
−γ
bi+ )fi+ = f0 ,
i = p + 1, . . . , s,
−
+
fs+1
= fs+1
= fs+1 .
Then there exists a non-zero vector π = (π0 , . . . , πl ), π0 ≤ 0 and a solution χ(t), ψ(t)
of the system (5) such that the condition a) and c) are fulfilled. Moreover,
e t = −ψ(e
πQ
t0 )[ϕ(
ė e
t0 )−
0
k
X
j=1
Aj (e
t0 )ϕ(η
ė j (e
t0 ))+f0 ]+
e t = −ψ(e
πQ
t1 )
0
s
X
i=p+1
X
k
j=1
ψ(γi )fi + χ(e
t0 )ϕ(
ė e
t0 ),
Aj (e
t1 )x(η
ė j (e
t1 )) + fs+1 .
Finally we note that the optimal control problems for various classes of delay and
neutral differential equations with discontinuous initial condition are considered in [1]–
[4].
References
1. G. Kharatishvili and T. Tadumadze, Nonlinear optimal control problem with
variable delays,non-fixed initial moment,and a piecewise-continuous prehistory. (Russian)
Tr. Mat. Inst. Steklova 220 (1998), 236–255; translation in Proc. Steklov Inst. Math.
220 (1998), 233–252.
2. T. Tadumadze, On new necessary condition of optimality of the initial moment
in control problems with delay. Mem. Differential Equations Math. Phys. 17 (1999),
157–159.
3. T. Tadumadze and L. Alkhazishvili, Necessary conditions of optimality for optimal problems with delays and with a discontinuous initial condition. Mem. Differential
Equations Math. Phys. 22 (2001),154–158.
4. N. Gorgodze, Necessary conditions of optimality in neutral type optimal problems
with non-fixed initial moment. Mem. Differential Equations Math. Phys. 19 (2000),
150–153.
Author’s address:
Department of Mathematics No. 99
Georgian Technical University
77, M. Kostava St., Tbilisi 0175
Georgia