Mem. Differential Equations Math. Phys. 20(2000), 136–140

T. Tadumadze and K. Gelashvili

AN EXISTENCE THEOREM FOR A CLASS OF OPTIMAL PROBLEMS WITH DELAYED ARGUMENT

(Reported on October 25, 1999)

1. Statement of the Problem. An Existence Theorem

Let J = [a, b] be a finite closed interval; O ⊂ ℝ^{n}–be an open set; K_{i}, i = 0, 1, U ⊂ ℝ^{r}, V ⊂ ℝ^{p} be compact sets; for each fixed (x_{1}, x_{2}, u_{1}, u_{2}) ∈ O^{2} × U^{2} let the function f : J × O^{2} × U^{2} → ℝ^{n} be measurable with respect to t ∈ J; for an arbitrary compact K ⊂ O there exist measurable functions m_{K}(t), L_{K}(t), t ∈ J, such that

|f(t, x_{1}, x_{2}, u_{1}, u_{2})| ≤ m_{K}(t), ∀(t, x_{1}, x_{2}, u_{1}, u_{2}) ∈ J × K^{2} × U^{2},

|f(t, x′_{1}, x′_{2}, u_{1}, u_{2}) − f(t, x′′_{1}, x′′_{2}, u_{1}, u_{2})|| ≤ L_{K}(t) ∑_{i=1}^{2} |x′′_{i} − x′_{i}|,

∀(t, x′_{1}, x′_{2}, x′′_{1}, x′′_{2}, u_{1}, u_{2}) ∈ J × K^{4} × U^{4}.

Further, let the functions τ(t), θ(t), t ∈ J, be absolutely continuous and satisfy the conditions: τ(t) ≤ t, τ̇(t) > 0, θ(t) ≤ t, θ̇(t) > 0; Ω = Ω(J_{0}, V, m, L) be the set of piecewise continuous functions v : J_{0} = [a_{0}, b_{0}] → V satisfying the condition: for each function v(·) ∈ Ω there exists a partition a_{0} = ξ_{0} < ⋯ < ξ_{n} = b_{0} such that the restriction of the function v(t) satisfies the Lipschitz condition on the open interval (ξ_{i}, ξ_{i+1}), i = 0, …, m, i.e., |v(t′) − v(t′′)| ≤ L|t′ − t′′|, ∀t′, t′′ ∈ (ξ_{i}, ξ_{i+1}), i = 0, …, m, where the numbers m and L do not depend on v ∈ Ω; let Ω_{0} = Ω([τ(a), b], K_{0}, m_{0}, L_{0}), elements of this set will be denoted by φ(·); Ω_{1} = Ω([θ(a), b], U, m_{1}, L_{1}), its elements being denoted by u(·); let q^{i} : j × O^{2} → ℝ^{1}, i = 0, …, l, be continuous functions.

Consider the problem:

ẋ(t) = f(t, x(t), x(τ(t)), u(t), u(θ(t))), t ∈ [t_{0}, t_{1}] ⊂ J, u(·) ∈ Ω_{1}  (1)
x(t) = φ(t), t ∈ [τ(t_{0}), t_{0}), x(t_{0}) = x_{0}, φ(·) ∈ Ω_{0}, x_{0} ∈ K_{1},  (2)
q^{i}(t_{0}, t_{1}, x_{0}, x(t_{1})) = 0, i = 0, …, l,  (3)
q^{0}(t_{0}, t_{1}, x_{0}, x(t_{1})) → min .  (4)

Definition 1. The function x(t) = x(t, z) ∈ O, t ∈ [τ(t_{0}), t_{1}] , is said to be a solution corresponding to the element z = (t_{0}, t_{1}, x_{0}, φ(·), u(·)) ∈ A = J^{2} × K_{1} × Ω_{0} × Ω_{1}, if on [τ(t_{0}), t_{0}] it satisfies the condition (2), while on the interval [t_{0}, t_{1}] it is absolutely continuous and the pair (u(·), x(·)) almost everywhere (a.e.) on [t_{0}, t_{1}] satisfies the equation (1).

Definition 2. The element z ∈ A is said to be admissible if the corresponding solution x(t) satisfies the condition (3).

The set of admissible elements will be denoted by Δ.

2000 Mathematics Subject Classification. 49J25.

Key words and phrases. Delay, existence of optimal control.

Definition 3. The element z̃ = (t̃_{0}, t̃_{1}, x̃_{0}, φ̃(·), ũ(·)) ∈ Δ is said to be optimal if

Ĩ = I(z̃) = inf_{z∈Δ} I(z),

where

I(z) = q^{0}(t_{0}, t_{1}, x_{0}, x(t_{1})), x(t) = x(t, z).

Theorem 1. Let the following conditions be valid:

1) Δ ≠ ∅;

2) there exists a compact set K_{2} ⊂ O such that

x(t, z) ∈ K_{2}, ∀z ∈ Δ.

Then there exists an optimal element.

2. Auxiliary Lemmas

Lemma 1. Let x_{k}(t) = x(t, z_{k}), t ∈ [τ(t_{0}^{k}), t_{1}^{k}], be the solution corresponding to the element z_{k} ∈ A; t_{0}^{k} → t_{0}, t_{1}^{k} → t_{1} as k → ∞, t_{0}^{k} ≥ t_{0}, t_{1}^{k} ≤ t_{1}; K_{i} ⊂ O, i = 3, 4, be compact sets with K_{3} ⊂ intK_{4} and x_{k}(t) ∈ K_{3}, t ∈ [t_{1}^{k}, t_{2}^{k}]. Then for sufficiently large k the functional differential equation

ẏ(t) = f(t, y(t), h(t_{0}^{k}, φ_{k}(·), y_{k}(·))(τ(t)), u_{k}(t), u_{k}(θ(t))),  (5)
y(t_{0}^{k}) = x_{0}^{k},

where

h(t_{0}, φ(·), y(·))(t) = \{ \begin{matrix}φ(t), & t ∈ [τ(a), t_{0}), \\ y(t), & t ∈ [t_{o}, b], \end{matrix}

has a solution y_{k}(t) = y(t, z_{k}) ∈ K_{4} defined on [t_{0}, t_{1}], and y_{k}(t) = x_{k}(t), t ∈ [t_{0}^{k}, t_{1}^{k}].

The proof of this lemma can be carried out in the standard way (for example, see Theorem 2 in [1]), since (5) is an ordinary differential eqation for t < t_{0}^{k}, and is a differential equation with delayed argument for t > t_{0}^{k}.

Lemma 2. Let v_{k}(·) ∈ Ω, k = 1, 2, . . .. Then there exists a subsequence of the sequence {v_{k}(·)}_{k=1}^{∞} such that it converges to some function v_{o}(·) ∈ Ω for each t ∈ J, except for not more than (m + 1) points.

Proof. By assumption the function v_{k}(t), t ∈ (ξ_{i}^{k}, ξ_{i+1}^{k}), satisfies Lipschitz condition. From this it imediately follows the existence of one-sided limits

lim_{t→ξ_{i}^{k}−} v_{k}(t) = v_{k_{i}}^{−}, i = 0, …, q − 1, lim_{t→ξ_{i}^{k}+} v_{k}(t) = v_{k_{i}}^{+}, i = 1, …, q.

We set the function

ω_{k_{i}}(t) = \{ \begin{matrix}v_{k_{i}}^{−}, & t ≤ ξ_{i}^{k}, \\ v_{k}(t), & t ∈ (ξ_{i}^{k}, ξ_{i+1}^{k}), \\ v_{k_{i}}^{+}, & t ≥ ξ_{i+1}^{k}, \end{matrix}

ω_{k}(t) = ∑_{i=0}^{m} χ_{k_{i}}(t)ω_{k_{i}}(t), t ∈ J_{0}, ω_{k}(b_{0}) = ω_{k}(b_{0}−),

where χ_{k_{i}}(t) is the characteristic function of the semi-open interval E_{k_{i}} = [ξ_{i}^{k}, ξ_{i+1}^{k}). Obviously, ω_{k}(·) ∈ Ω and

ω_{k}(t) = v_{k}(t), t ∈ (ξ_{i}^{k}, ξ_{i+1}^{k}).  (6)

The sequence {ω_{k_{i}}(t)}_{k=1}^{∞} is uniformly bounded and equicontinuous for each i = 0, …, m. Therefore, by virtue of Arzela-Ascoli’s lemma, from {ω_{k_{i}}(t)}_{k=1}^{∞} it can be picked out a uniformly convergent subsequence which again is denoted by {ω_{k_{i}}(t)}_{k=1}^{∞} .

Thus

lim_{k→∞} ω_{k_{i}}(t) = ω_{i}(t) uniformly for t ∈ J_{0}.

Without loss of generality we will assume that

lim_{k→∞} ξ_{i}^{k} = ξ_{i}, i = 1, …, q − 1.

Conseqently we have

lim_{k→∞} E_{k_{i}} = E_{i}, lim_{k→∞} χ_{k_{i}}(t) = χ_{i}(t), t ∈ ℝ,

where E_{i} is an interval and χ_{i}(t) is the charasteristic function of the interval E_{i}. Therefore for each t ∈ J_{0}

lim_{k→∞} ω_{k}(t) = ω(t) = ∑_{i=0}^{m} χ_{i}(t)ω_{i}(t),

besides ω(·) ∈ Ω.

Taking into account (6), we can conclude that

lim_{k→∞} v_{k}(t) = ω(t) = v_{0}(t), t ∈ (ξ_{i}, ξ_{i+1}), i = 0, 1, …, m. □

3. Proof of the Theorem

There exists a sequence z_{k} = (t_{0}^{k}, t_{1}^{k}, x_{0}^{k}, φ_{k}(·), u_{k}(·)) ∈ Δ, k = 1, 2, …, such that

I(z_{k}) → Ĩ, t_{0}^{k} → t̃_{0}, t_{1}^{k} → t̃_{1}, x_{0}^{k} → x̃_{0} as k → ∞;
lim_{k→∞} φ_{k}(t) = φ̃(t), a.e. on [τ(a), b], φ̃(·) ∈ Ω_{0};
lim_{k→∞} u_{k}(t) = ũ(t), a.e. on [θ(a), b], ũ(·) ∈ Ω_{1}

(see Lemma 2).

Consider the case where t_{0}^{k} ≥ t̃_{0}, t_{1}^{k} ≤ t̃_{1}. The remaining cases can be considered analogously.

Let K_{5} ∈ O be a compact set, K_{2} ∈ intK_{5}. For sufficiently large k ≥ k_{0} there exists the solution y_{k}(t) ∈ K_{5} of the equation (5) defined on [t̃_{0}, t̃_{1}] and y_{k}(t) = x_{k}(t), t ∈ [t_{0}^{k}, t_{1}^{k}], (see Lemma 1).

Obviously

h(t_{0}^{k}, φ_{k}(·), y_{k}(·))(t) ∈ K_{6}, k ≥ k_{0}, t ∈ [τ(t̃_{0}), t̃_{1}], K_{6} = K_{5} ∪ K_{0},

therefore

|ẏ(t)| ≤ m_{K_{6}}(t), t ∈ [t̃_{0}, t̃_{1}].

Thus the sequence {y_{k}(·)}_{k=1}^{∞} is uniformly bounded and equicontinouos. Without loss of generality we can assume that

lim_{k→∞} y_{k}(t) = ỹ(t) uniformly with t ∈ [t̃_{0}, t̃_{1}].

Consequently,

lim_{k→∞} f_{k}[t] = f̃[t], a.e. t ∈ [t̃_{0}, t̃_{1}],

where

f_{k}[t] = f(t, y_{k}(t), h(t_{0}^{k}, φ_{k}(·), y_{k}(·))(τ(t)), u_{k}(t), u_{k}(θ(t))),
f̃[t] = f(t, y(t), h(t̃_{0}, φ̃(·), ỹ(·))(τ(t)), ũ(t), ũ(θ(t))).

Further,

y_{k}(t) = x_{0}^{k} + ∫_{t_{0}^{k}}^{t} f̃[s]ds + α_{k} + β_{k}(t),  (7)

where

α_{k} = ∫_{t̃_{0}}^{t_{0}^{k}} f_{k}(t]dt, β_{k}(t) = ∫_{t̃_{0}}^{t} [f_{k}[s] − f̃[s]]ds.

Evidently

lim_{k→∞} α_{k} = 0, |β_{k}(t)| ≤ ∫_{t̃_{0}}^{t̃_{1}} |f_{k}[s] − f̃[s]|ds.

By virtue of Lebesgue’s theorem on passage to limit under the integral sign we have

lim_{k→∞} β_{k}(t) = 0 uniformly with t ∈ [t̃_{0}, t̃_{1}].

From (7) as k → ∞ we get

ỹ(t) = x̃_{0} + ∫_{t̃_{0}}^{t} f̃[s]ds.

It is easy to see that

lim_{k→∞} y_{k}(t_{1}^{k}) = ỹ(t̃_{1}),

therefore

q^{i}(t̃_{0}, t̃_{1}, x̃_{0}, ỹ(t̃_{1})) = 0, i = 1, …, l, Ĩ = q^{0}(t̃_{0}, t̃_{1}, x̃_{0}, ỹ(t̃_{1})).

Introduce the function

x̃(t) = \{ \begin{matrix}φ̃(t), & t ∈ [τ(t̃_{0}), t̃_{0}), \\ ỹ(t), & t ∈ [t̃_{0}, t̃_{1}). \end{matrix}

Obviously z̃ = (t̃_{0}, t̃_{1}, x̃_{0}, x̃(·)) ∈ Δ and I(z̃) = Ĩ.

Finally, note that the proved theorem is also valid in the case where the right-hand side of the equation (1) has the form

f(t, x(τ_{1}(t)), …, x(τ_{s}(t)), u(θ_{1}(t)), …, u(θ_{ν}(t))),

where the functions τ_{i}(t), i = 1, …, s, θ_{i}(t), i = 1, …, ν, are absolutely continuous and satisfy the conditions τ_{i}(t) ≤ t, τ̇_{i}(t) > 0; θ_{i}(t) ≤ t, θ̇_{i}(t) > 0.

If K_{0}, U are convex sets and the points of discontinuity of the functions from the set Ω_{i}, i = 0, 1, are fixed be forehand, then for the problem (1)–(4) necessary conditions of optimality are valid in the form given in [2]. In the class of measurable functions the problem of existence is studied in [3,4].

References

1. G. Kharatishvili, T Tadumadze, and N. Gorgodze, Continuous dependence and differentiability of solution with respect to initial data and right-hand side for differential equations with deviating argument. Mem. Differential Equations Math. Phys. 19(2000), 3–105.

2. G. Kharatishvili and T Tadumadze, The maximum principle in optimal control problems with concentrated and distributed delays in controls. Georgian Math. J. l2(1995), No. 6, 577–591.

3. T Tadumadze, Existence theorem for solution of optimal problems with variable delays. Control and cybernetics. 10(1981), Nos. 3–4, 125–134.

4. T Tadumadze, The maximum principle and existence theorem in the optimal problems with delay and non-fixed initial function. Seminar of I. Vekua Institute of Applied Mathematics. Reports 22(1993), 102–107.

Authors’ addresess:

T. Tadumadze I. Vekua Institute of Applied Mathematics I. Javakhishvili Tbilisi State University 2, University St., Tbilisi 380043 Georgia

K. Gelashvili Department of Applied Mathematics and Computer Sciences I. Javakhishvili Tbilisi State University 2, University St., Tbilisi 380043 Georgia