Mem. Differential Equations Math. Phys. 17(1999), 157–159

T. Tadumadze

ON NEW NECESSARY CONDITION OF OPTIMALITY OF THE INITIAL MOMENT IN CONTROL PROBLEMS WITH DELAY

(Reported on November 9, 1998)

Necessary conditions of optimality are obtained for optimal control problems with non fixed initial moment. The condition at the optimal initial moment, unlike the early known condition [1], contains a new term.

Let O ⊂ R^{n}, G ⊂ R^{r} be open sets, J = [a, b] be a finite interval and let the function f : J × O^{2} × G^{2} → R^{n} satisfies the following conditions:

1) for a fixed t ∈ J the function f(t, x_{1}, x_{2}, u_{1}, u_{2}) is continuous with respect to (x_{1}, x_{2}, u_{1}, u_{2}) ∈ O^{2} × G^{2} and continuously differentiable with respect to (x_{1}, x_{2}) ∈ O^{2};

2) for a fixed (x_{1}, x_{2}, u_{1}, u_{2}) ∈ O^{2} × G^{2} the functions f, f_{x_{i}}, i = 1, 2 are measurable with respect to t; for arbitrary compacts K ⊂ O, V ⊂ G there exists a function m_{K,V} (·) ∈ L_{1}(J, R_{0}^{+}), R_{0}^{+} = [0, ∞), such that

|f(t, x_{1}, x_{2}, u_{1}, u_{2})| + ∑_{i=1}^{2} |f_{x_{i}}(·)| ≤ m_{K,V} (t), ∀(t, x_{1}, x_{2}, u_{1}, u_{2}) ∈ J × K^{2} × V^{2}.

The following asumptions will be made: τ : R^{1} → R^{1} and θ : R^{1} → R^{1} are absolutely continuous functions satisfying τ(t) ≤ t, τ̇(t) > 0, θ(t) < t, θ̇(t) > 0 for t ∈ R^{1}; Δ is the set of piecewise continuous functions φ : [τ(a), b] → N with a finite number of discontinuity points with N ⊂ O convex bounded set, ∥φ∥ = sup{|φ| : t ∈ J_{1}}; Ω is the set of measurable functions, u : [θ(a), b] → U is such that cl{u(t) : t ∈ [θ(a), b]} is a compact lying in G, U ⊂ G is an arbitrary set; q^{i} : J^{2} × O^{2} → R^{n}, i = 0, …, l, are continuously differentiable functions; γ(t) is the function inverse to τ(t).

To every element σ = (t_{0}, t_{1}, x_{0}, φ, u) ∈ A = J × O × Δ × Ω, t_{0} < t_{1} there corresponds the differential equation

ẋ(t) = f(t, x(t), x(τ(t)), u(t), u(θ(t))), t ∈ [t_{0}, t_{1}],  (1)

with the initial condition

x(t) = φ(t), t ∈ [τ(t_{0}), t_{0}), x(t_{0}) = x_{0}.  (2)

Definition 1. The function x(t) = x(t, σ) ∈ O, t ∈ [τ(t_{0}), t_{1}], t_{0} ∈ [a, t_{1}), is said to be a solution corresponding to the element σ ∈ A if on [τ(t_{0}), t_{0}] it satisfies the condition (2), while on the interval [t_{0}, t_{1}] is absolutely continuous and almost everywhere satisfies the equation (1).

Definition 2. The element σ ∈ A is said to be admissible if the corresponding solution x(t) satisfies q^{i}(t_{0}, t_{1}, x_{0}, x(t_{1})) = 0, i = 1, …, l2.

The set of admissible elements will denoted by A_{0}.

Definition 3. The element σ = (t̃_{0}, t̃_{1}, x̃_{0}, φ̃, ũ) ∈ A_{0} is said to be locally optimal if there exist a number δ > 0 and a compact set K ⊂ O such that for an arbitrary element

1991 Mathematics Subject Classification. 49K25.

Key words and phrases. Delay, necessary condition of optimality. σ ∈ A_{0} satisfying |t̃_{0} − t_{0}| + |t̃_{1} − t_{1}| + |x̃_{0} − x_{0}| + ∥φ̃ − φ∥ + ∥f̃− f∥_{K} ≤ δ the inequality q^{0}(t̃_{0}, t̃_{1}, x̃_{0}, x̃(t̃_{1})) ≤ q^{0}(t_{0}, t_{1}, x_{0}, x(t_{1})) holds.

Here

\begin{matrix}∥f̃− f∥_{K} = ∫_{J} H(t; f, K)dt, \\ H(t; f, K) = sup \{|f̃(t, x_{1}, x_{2}) − f(t, x_{1}, x_{2})| + ∑_{i=1}^{2} |f̃_{x_{i}}(·) − f_{x_{i}}(·)| : (x_{1}, x_{2}) ∈ K^{2}\}; \end{matrix}

f̃(t, x_{1}, x_{2})=f(t, x_{1}, x_{2}, ũ(t), ũ(θ(t))), f(t, x_{1}, x_{2})=f(t, x_{1}, x_{2}, u(t), u(θ(t))), x̃(t)=x(t, σ̃). The problem of optimal control consists in finding a locally optimal element. Theorem 1. Let σ̃ ∈ A_{0}, t̃_{0} ∈ (a, b), t̃_{1} ∈ (a, b], γ_{0} = γ(t̃_{0}) ∈ [t̃_{0}, t̃_{1}) be a locally

optimal element and there exist the finite limits

lim_{ω→ω_{0}^{−}} f̃(ω) = f_{0}^{−}, ω ∈ R− ; t̃_{0} × O^{2}, R− ; t̃_{0} = (−∞, t̃_{0}], ω_{0}^{−} = (t̃_{0}, x̃_{0}, φ̃(τ(t̃_{0}^{−}))),

\begin{matrix}lim_{(ω_{1},ω_{2})→(ω_{1}^{0},ω_{2}^{−})}[f̃(ω_{1}) − f̃(ω_{2})] = f_{1}^{−}, ω_{i} ∈ R− ; γ̃_{0} × O^{2}, i = 1, 2, ω_{1}^{0} = (γ_{0}, x̃(γ_{0}), x̃_{0}), \\ ω_{2}^{−} = (γ_{0}, x̃(γ_{0}), φ̃(t̃_{0}^{−})), lim_{t→t̃_{0}^{−}} γ̇(t) = γ̇^{−}, \end{matrix}

lim_{ω→ω_{3}^{−}} f̃(ω) = f_{2}^{−}, ω ∈ R− ; t̃_{1} × O^{2}, ω_{3}^{−} = (t̃_{1}, x̃(t̃_{1}), x̃(τ(t̃_{1}^{−}))).  (3)

Then there exist a non-zero vector π = (π_{0}, …, π_{l}), π_{0} ≤ 0 and a solution ψ(t), t ∈ [t̃_{0}, γ(t̃_{1})] of the adjoint equation

ψ̇(t) = −ψ(t)f̃_{x_{1}}[t] − ψ(γ(t))f̃_{x_{2}}[γ(t)]γ̇(t), t ∈ [t̃_{0}, t̃_{1}], ψ(t) = 0, t ∈ (t̃_{1}, γ(t̃_{1})], (4)

such that the following conditions are fulfilled:

∫ t̃_{0} ; τ(t̃_{0})ψ(γ(t))f̃_{x_{2}}[γ(t)]γ̇(t)φ̃(t)dt ≥ ∫ t̃_{0} ; τ(t̃_{0})ψ(γ(t))f̃_{x_{2}}[γ(t)]γ̇(t)φ(t)dt, ∀φ ∈ Δ, (5)
∫ t̃_{1} ; t̃_{0}ψ(t)f̃[t]dt ≥ ∫ t̃_{1} ; t̃_{0}ψ(t)f(t, x̃(t), x̃(τ(t)), u(t), u(θ(t)))dt, ∀u ∈ Ω;  (6)
πQ̃_{x_{0}} = −ψ(t̃_{0}), πQ̃_{x_{1}} = ψ(t̃_{1}),  (7)
πQ̃_{t_{0}} ≥ ψ(t̃_{0})f_{0}^{−} + ψ(γ_{0})f_{1}^{−}γ_{0}^{−}, πQ̃_{t_{1}} ≥ −ψ(t̃_{1})f_{2}^{−}.  (8)

Here Q = (q^{0}, …, q^{l}), the tilde over Q means that there corresponding gradient is calculated at the point (t̃_{0}, t̃_{1}, x̃_{0}, x̃(t̃_{1})); f̃_{x_{i}}[t]=f̃_{x_{i}}(t, x̃(t), x̃(τ(t))), f̃[t]=f̃(t, x̃(t), x̃(τ(t))).

If rank (Q̃_{x_{0}}, Q̃_{x_{1}}) = 1 + l, then in the theorem 1 ψ(t) ≢ 0.

Theorem 2. Let σ̃ ∈ A_{0}, t̃_{0} ∈ [a, b), t̃_{1} ∈ (a, b), γ_{0} ∈ [t̃_{0}, t̃_{1}) be a locally optimal element and there exist the finite limits

lim_{ω→ω_{0}^{+}} f̃(ω) = f_{0}^{+}, ω ∈ R+ ; t̃_{0} × O^{2}, ω_{0}^{+} = (t̃_{0}, x̃_{0}, φ̃(τ(t̃_{0}^{+}))),

\begin{matrix}lim_{(ω_{1},ω_{2})→(ω_{1}^{0},ω_{2}^{+})}[f̃(ω_{1}) − f̃(ω_{2})]=f_{1}^{+}, ω_{i} ∈R+ ; γ̃_{0} × O^{2}, i=1, 2, ω_{2}^{+} =(γ_{0}, x̃(γ_{0}), φ̃(t̃_{0}^{+})), \\ lim_{t→t̃_{0}^{+}} γ̇(t) = γ̇^{+}, \end{matrix}

lim_{ω→ω_{3}^{+}} f̃(ω) = f_{2}^{+}, ω ∈ R+ ; t̃_{1} × O^{2}, ω_{3}^{+} = (t̃_{1}, x̃(t̃_{1}), x̃(τ(t̃_{1}^{+}))).  (9)

Then there exist a non-zero vector π = (π_{0}, …, π_{l}), π_{0} ≤ 0 and a solution ψ(t), t ∈ [t̃_{0}, γ(t̃_{1})] of the equation (4) such that the conditions (5)–(7) are fulfilled. Moreover,

πQ̃_{t_{0}} ≤ ψ(t̃_{0})f_{0}^{+} + ψ(γ_{0})f_{1}^{+}γ_{0}^{+}, πQ̃_{t_{1}} ≤ −ψ(t̃_{1})f_{2}^{+}.  (10)

Theorem 3. Let σ̃ ∈ A_{0}, t̃_{i} ∈ (a, b), i = 0, 1, γ_{0} ∈ (t̃_{0}, t̃_{1}) be a locally optimal element and the assumptions of Theorems 1, 2 are fulfilled. Let, besides,

f_{0}^{+} = f_{0}^{−} = f_{0}, f_{1}^{+}γ̇^{+} = f_{1}^{−}γ̇^{−} = f_{1}, f_{2}^{+} = f_{2}^{−} = f_{2}.  (11)

Then there exist a non-zero vector π = (π_{0}, …, π_{l}), π_{0} ≤ 0 and a solution ψ(t), t ∈ [t̃_{0}, γ(t̃_{1})] of the equation (4) such that the conditions (5)–(7) hold. Moreover,

πQ̃_{t_{0}} = ψ(t̃_{0})f_{0} + ψ(γ_{0})f_{1}, πQ̃_{t_{1}} = −ψ(t̃_{1})f_{2}.  (12)

If rank (Q̃_{t_{0}}, Q̃_{t_{1}}, Q̃_{x_{0}}, Q̃_{x_{1}}) = 1 + l, then in Theorem 3 ψ(t) ≢ 0.

Remark 1. Assume that the function (γ̇(t), φ̃(t), φ̃(τ(t))) is continuous at point t̃_{0} and the function f̃(t, x_{1}, x_{2}) is continuous at the points (t̃_{0}, x̃_{0}, φ̃(τ(t̃_{0}))), (γ_{0}, x̃(γ_{0}), x̃_{0}), (γ_{0}, x̃(γ_{0}), φ̃(t_{0})), (t̃_{1}, x̃(t̃_{1}), x̃(τ(t̃_{1}))), then, it is clear that in Theorem 3

\begin{matrix}f_{0} = f̃(t̃_{0}, x̃_{0}, φ̃(τ(t̃_{0}))), f_{1} = [f̃(γ_{0}, x̃(γ_{0}), x̃_{0}) − f̃(γ_{0}, x̃(γ_{0}), φ̃(t_{0}))]γ̇(t̃_{0}), \\ f_{2} = f̃(t̃_{1}, x̃(t̃_{1}), x̃(τ(t̃_{1}))). \end{matrix}

Theorem 4. Let σ̃ ∈ A_{0}, t̃_{i} ∈ (a, b), i = 0, 1, γ_{0} = t̃_{0} be a locally optimal element and the conditions of theorems 1, 2 and (11) are fulfilled. Moreover f_{0}^{−} + f_{1}^{−}γ̇^{−} = f_{0}^{+} + f_{1}^{+}γ̇^{+} = f̂_{1}. Then there exist a non-zero vector π = (π_{0}, …, π_{l}), π_{0} ≤ 0 and a solution ψ(t), t ∈ [t̃_{0}, γ(t̃_{1})] of the equation (4) such that the conditions (5), (7), (12) hold. Let, besides,

πQ̃_{t_{0}} = ψ(t̃_{0})f̂_{1}.

Theorem 5. Let σ̃ ∈ A_{0}, t̃_{i} ∈ (a, b), i = 0, 1, γ_{0} = t̃_{0} be a locally optimal element, the condition (11) hold, the function (γ̇(t), φ̃(t)) be continuous at the point t̃_{0} and

f_{3}^{−} = f_{3}^{+} = f_{3},  (13)

where

lim_{ω→ω_{4}} f̃(ω) = f_{3}^{−}, ω ∈ R− ; t̃_{0} × O^{2}, ω_{4} = (t̃_{0}, x̃_{0}, x̃_{0}), lim_{ω→ω_{4}} f̃(ω) = f_{3}^{+}, ω ∈ R+ ; t̃_{0} × O^{2}.

Let, besides, there exist a neighborhood V (ω_{5}) of the point ω_{5} = (t̃_{0}, x̃_{0}, φ̃(t̃_{0}) such that the function f̃(ω), ω ∈ V (ω_{5}) is bounded. Then there exist a non-zero vector π = (π_{0}, …, π_{l}), π_{0} ≤ 0 and a solution ψ(t), t ∈ [t̃_{0}, γ(t̃_{1})] of the equation (4) such that the conditions (6), (7), (12) hold. Moreover,

πQ̃_{t_{0}} = ψ(t̃_{0})f_{3}.  (14)

Theorem 6. Let σ̃ ∈ A_{0}, t̃_{i} ∈ (a, b), i = 0, 1, be a locally optimal element. The conditions (11), (13) hold. Let, besides, in some neighborhood of t̃_{0} the function τ(t) = t. Then there exist a non-zero vector π = (π_{0}, …, π_{l}), π_{0} ≤ 0 and a solution ψ̃(t), t ∈ [t̃_{0}, γ(t̃_{1})] of the equation (4) such that the conditions (6), (7), (12), (14) hold.

References

1. 1. G. L. Kharatishvili, A maximum principle in extremal problems with delays. In: Mathematical theory of control, Academic press (1967), 26–34.

Author’s address:

I. Vekua Institute of Applied Mathematics

Tbilisi State University

2, University St., Tbilisi 380043

Georgia