Mem. Differential Equations Math. Phys. 19(2000), 150–153

N. Gorgodze

NECESSARY CONDITIONS OF OPTIMALITY IN NEUTRAL TYPE OPTIMAL PROBLEMS WITH NON-FIXED INITIAL MOMENT

(Reported on April 22, 1999)

Let J = [a, b] be a finite interval; O ⊂ ℝ^{n}, G ⊂ ℝ^{r} be open sets; τ : ℝ^{1} → ℝ^{1}, η : ℝ^{1} → ℝ^{1} be absolutely continuous and continuously differentiable functions, respectively, satisfying the conditions: τ(t) ≤ t, τ̇(t) > 0, η(t) < t, η̇(t) > 0; γ(t) = τ^{−1}(t), σ(t) = η^{−1}(t); q^{i} : J^{2} × O^{2} → ℝ^{1}, i = 0, …, l, be continuously differentiable functions; Δ = Δ(J_{1}, M) be the set of continuously differentiable functions φ : J_{1} → M, J_{1} = [ρ(a), b], ρ(t) = min{η(t), τ(t)}, t ∈ J, ∥φ∥ = sup{|φ(a)| + |φ̇(t)| : t ∈ J_{1}}, M ⊂ O be a convex bounded set; Ω_{1} be the set of measurable functions u : J → U such that cl{u(t) : t ∈ J} ⊂ G is compact, U ⊂ G be an arbitrary set; Ω_{2} be a set of measurable functions v : J → V, V ⊂ G be a convex bounded set; A(t, v) the an n × n dimensional matrix function, continuous on J × V and continuously differentiable with respect to v ∈ V.

Next, let the function f : J × O^{2} × G → ℝ^{n} satisfy the following conditions:

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

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

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

To every element μ = (t_{0}, t_{1}, x_{0}, φ, u, v) ∈ B = J^{2} × O × Δ × Ω_{1} × Ω_{2}, t_{0} < t_{1}, there corresponds the differential equation

ẋ(t) = A(t, v(t))ẋ(η(t)) + f(t, x(t), x(τ(t)), u(t)), t ∈ [t_{0}, t_{1}],  (1)

with the initial condition

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

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

Definition 2. The element μ ∈ B is said to be admissible, if the corresponding solution x(t) satisfies the conditions

q^{i}(t_{0}, t_{1}, x_{0}, x(t_{1})) = 0, i = 1, …, l .

1991 Mathematics Subject Classification. 49K25.

Key words and phrases. Neutral type equation, necessary condition of optimality.

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

|t̃_{0} − t_{0}| + |t̃_{1} − t_{1}| + |x̃_{0} − x_{0}| + ∥φ̃ − φ∥ + ∥f̃− f∥_{X} + sup_{t∈J}|ṽ(t) − v(t)| ≤ 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}))

is fulfilled.

Here

\begin{matrix}∥f̃− f∥_{X} = ∫_{J} H(t; f, X), \\ H(t; f, X) = 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}) ∈ X^{2}\}; \end{matrix}

f̃(t, x_{1}, x_{2}) = f(t, x_{1}, x_{2}, ũ(t)), f(t, x_{1}, x_{2}) = f(t, x_{1}, x_{2}, u(t)), x̃(t) = x(t, μ̃). The problem of optimal control consists in finding a locally optimal element.

Theorem 1. Let μ̃ ∈ B_{0}, t̃_{i} ∈ (a, b), i = 0, 1, be a locally optimal element, ṽ(t) be a piecewise continuous function; γ_{0} = γ(t̃_{0}) ∈ (t̃_{0}, t̃_{1}), σ_{0} = σ(t̃_{0}) ∈ (t̃_{0}, t̃_{1}), there exist integer numbers m_{i} ≥ 0, i = 1, 2, such that γ_{0} ∈ (η^{m_{1}+1}(t̃_{1}), η^{m_{1}}(t̃_{1})), σ_{0} ∈ (η^{m_{2}+1}(t̃_{1}), η^{m_{2}}(t̃_{1})) (η^{i}(t) = η(η^{i−1}(t)), η^{0}(t) = t) and there exist the finite limits:

lim_{ω→ν_{0}}f̃(ω) = f_{0}^{−}, ω = (t, x_{1}, x_{2}) ∈ ℝ− ; t̃_{0} × O^{2}, ℝ− ; t̃_{0} = (−∞, t̃_{0}], ν_{0} = (t̃_{0}, x̃_{0}, φ̃(τ(t̃_{0})));

lim_{(ω_{1},ω_{2})→(ν_{1},ν_{2})}[f̃(ω_{1}) − f̃(ω_{2})] = f_{1}^{−}, ω_{i} ∈ ℝ_{γ_{0}}^{−} × O^{2}, i = 1, 2, ν_{1} = (γ_{0}, x̃(γ_{0}), x̃_{0});

ν_{2} = (γ_{0}, x̃(γ_{0}), φ̃(t̃_{0})), lim_{t→t̃_{0}^{−}}γ̇(t) = γ̇^{−};

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

lim_{t→t̃_{i}^{−}}Ã(t) = A− ; t̃_{i}, i = 0, 1, Ã(t) = A(t, ṽ(t));

lim_{t→σ^{i}(γ_{0})}Ã(t) = A_{σ^{i}(γ_{0})}^{−}, t ∈ ℝ_{σ^{i}(γ_{0})}^{−}, i = 1, …, m_{1} ;

lim_{t→σ^{i}(σ_{0})}Ã(t) = A_{σ^{i}(σ_{0})}^{−}, t ∈ ℝ_{σ^{i}(σ_{0})}^{−}, i = 0, …, m_{2} ;

Then there exists a non-zero vector π = (π_{0}, …, π_{l}), π_{0} ≤ 0, and solutions ψ(t), χ(t) of the system

\{ \begin{matrix}χ̇(t) = −ψ(t)f̃_{x_{1}}[t] − ψ(γ(t))f̃_{x_{2}}[γ(t)]γ̇(t), \\ ψ(t) = χ(t) + ψ(σ(t))Ã(σ(t))σ̇(t), t ∈ [t̃_{0}, t̃_{1}], ψ(t) = 0, t > t̃_{1}, \end{matrix}  (3)

such that the following conditions are fulfilled:

∫_{τ(t̃_{0})}^{t̃_{0}} ψ(γ(t))f̃_{x_{2}}[γ(t)]γ̇(t)φ̃(t)dt + ∫_{η(t̃_{0})}^{t̃_{0}} ψ(σ(t))Ã(σ(t))σ̇(t) ˙φ̃(t)dt ≥

≥ ∫_{τ(t̃_{0})}^{t̃_{0}} ψ(γ(t))f̃_{x_{2}}[γ(t)]γ̇(t)φ(t)dt + ∫_{η(t̃_{0})}^{t̃_{0}} ψ(σ(t))Ã(σ(t))σ̇(t)φ̇(t)dt, ∀φ ∈ Δ, (4)

∫_{t̃_{0}}^{t̃_{1}} ψ(t)f̃[t]dt ≥ ∫_{t̃_{0}}^{t̃_{1}} ψ(t)f(t, x̃(t), x̃(τ(t)), u(t))dt, ∀u ∈ Ω_{1};  (5)

∫_{t̃_{0}}^{t̃_{1}} ψ(t)Ã_{v}(t) ˙x̃(η(t))ṽ(t)dt ≥ ∫_{t̃_{0}}^{t̃_{1}} ψ(t)Ã_{v}(t) ˙x̃(η(t))v(t)dt, ∀v ∈ Ω_{2};  (6)

πQ̃_{x_{0}} = −χ(t̃_{0}), πQ̃_{x_{1}} = χ(t̃_{1}),  (7)

πQ̃_{t_{0}} ≥ χ(t̃_{0})[A− ; t̃_{0} ˙φ̃(η(t̃_{0})) +

+f_{0}^{−}] + ψ(σ_{0}^{−})A_{σ_{0}}^{−}[A− ; t̃_{0} ˙φ̃(η(t̃_{0})) + f_{0}^{−} − ˙ φ̃(t̃_{0})]σ̇(t̃_{0}) + ψ(γ_{0}^{−})f_{1}^{−}γ̇^{−},

πQ̃_{t_{1}} ≥ −ψ(t̃_{1})[A− ; t̃_{1} ˙x̃(η(t̃_{1}^{−})) + f_{2}^{−}].

Here Q = (q^{0}, …, q^{l})^{T}, the tilde over Q means that the 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))).

Theorem 2. Let μ̃ ∈ B_{0}, t̃_{i} ∈ (a, b), i = 0, 1, be a locally optimal element, ṽ(t) be a piecewise continuous function; γ_{0} ∈ (t̃_{0}, t̃_{1}), σ_{0} ∈ (t̃_{0}, t̃_{1}), there exist integer numbers m_{i} ≥ 0, i = 1, 2, such that γ_{0} ∈ (η^{m_{1}+1}(t̃_{1}), η^{m_{1}}(t̃_{1})), σ_{0} ∈ (η^{m_{2}+1}(t̃_{1}), η^{m_{2}}(t̃_{1})) and there exist the finite limits:

lim_{ω→ν_{0}}f̃(ω) = f_{0}^{+}, ω ∈ ℝ+ ; t̃_{0} × O^{2}, lim_{t→t̃_{i}^{+}}Ã(t) = A+ ; t̃_{i}, i = 0, 1, lim_{t→t̃_{0}^{+}}γ̇(t) = γ̇^{+};

lim_{(ω_{1},ω_{2})→(ν_{1},ν_{2})}[f̃(ω_{1}) − f̃(ω_{2})] = f_{1}^{+}, ω_{i} ∈ ℝ_{γ_{0}}^{+} × O^{2}, i = 1, 2,

lim_{ω→ν_{3}}f̃(ω) = f_{2}^{+}, ω ∈ ℝ+ ; t̃_{1} × O^{2};

lim_{t→σ^{i}(γ_{0})}Ã(t) = A_{σ^{i}(γ_{0})}^{+}, t ∈ ℝ_{σ^{i}(γ_{0})}^{+}, i = 1, …, m_{1} ;

lim_{t→σ^{i}(σ_{0})}Ã(t) = A_{σ^{i}(σ_{0})}^{+}, t ∈ ℝ_{σ^{i}(σ_{0})}^{+}, i = 0, …, m_{2} .

Then there exists a non-zero vector π = (π_{0}, …, π_{l}), π_{0} ≤ 0, and solutions ψ(t), χ(t) of the system (3) such that the conditions (4) − (7) are fulfilled. Moreover,

πQ̃_{t_{0}} ≤ χ(t̃_{0})[A+ ; t̃_{0} ˙φ̃(η(t̃_{0}))+f_{0}^{+}]+ψ(σ_{0}^{+})A_{σ_{0}}^{+}[A+ ; t̃_{0} ˙φ̃(η(t̃_{0}))+f_{0}^{+}− ˙φ̃(t̃_{0})]σ̇(t̃_{0})+ψ(γ_{0}^{+})f_{1}^{+}γ̇^{+},

πQ̃_{t_{1}} ≤ −ψ(t̃_{1})[A+ ; t̃_{1} ˙x̃(η(t̃_{1}^{+})) + f_{2}^{+}].

Theorem 3. Let μ̃ ∈ B_{0}, t̃_{i} ∈ (a, b), i = 0, 1, be a locally optimal element, ṽ(t) be a piecewise continuous function; γ_{0} ∈ (t̃_{0}, t̃_{1}), σ_{0} ∈ (t̃_{0}, t̃_{1}), there exist integer numbers m_{i} ≥ 0, i = 1, 2, such that γ_{0} ∈ (η^{m_{1}+1}(t̃_{1}), η^{m_{1}}(t̃_{1})), σ_{0} ∈ (η^{m_{2}+1}(t̃_{1}), η^{m_{2}}(t̃_{1})) the function τ̇(t) be continuous at the point t̃_{0}, the function f̃(ω) be continuous at the points ν_{i}, i = 0, 1, 2, 3, the function Ã(t) be continuous at the points t̃_{0}, t̃_{1}, σ^{i}(γ_{0}), i = 1, …, m_{1} , σ^{i}(σ_{0}), i = 0, …, m_{2}, the function ˙ x̃(η(t)) be continuous at the point t̃_{1}.

Then there exists a non-zero vector π = (π_{0}, …, π_{l}), π_{0} ≤ 0, and solutions ψ(t), χ(t) of the system (3) such that the conditions (4)–(7) are fulfilled. Moreover,

πQ̃_{t_{0}} = χ(t̃_{0})[Ã(t̃_{0}) ˙φ̃(η(t̃_{0})) + f̃(ν_{0})] + ψ(σ_{0})Ã(σ_{0})[Ã(t̃_{0}) ˙φ̃(η(t̃_{0})) + f(ν_{0})−

− ˙φ̃(t̃_{0})]σ̇(t̃_{0}) + ψ(γ_{0})[f̃(ν_{1}) − f̃(ν_{2})]γ̇(t̃_{0}),
πQ̃_{t_{1}} = −ψ(t̃_{1})[A(t̃_{1}) ˙x̃(η(t̃_{1})) + f̃(ν_{3})].

Finally we note that the theorems formulated above are analogues of the theorems given in [1]. These theorems are proved using formulas for the differential of the solution with respect to the initial data and the right-hand side given in [2], by the scheme described in [3].

References

1. T. Tadumadze, On new necessary condition of optimality of initial moment in control problems with delay. Mem. Differential Equations Math. Phys. 17(1999), 157–159.

2. T. Tadumadze and N. Gorgodze, Differentiability of solution of differential equations with deviating argument with respect to the initial data and the right-hand side. Seminar of I. Vekua Institute of Applied Mathematics. Reports. 23(1997), 108–119.

3. R. Gamkrelidze and G. Kharatishvili, Extremal problems in linear topological spaces. (Russian) Izv. Akad. Nauk SSSR, Ser. Mat. 33(1969), No. 4, 781–839.

Author’s address:

I. Vekua Institute of Applied Mathematics

Tbilisi State University

2, University St., Tbilisi 380043

Georgia