EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A CLASS OF INFINITE-HORIZON SYSTEMS DERIVED FROM OPTIMAL CONTROL

This paper deals with the existence and uniqueness of solutions for a class of infinite-horizon systems derived from optimal control. An existence and uniqueness theorem is proved for such Hamiltonian systems under some natural assumptions.


Introduction
We begin with a simple example to introduce the background of the considered problem.Let U be a bounded closed subset of R m and let functions f : a,∞) → R be differentiable with respect to the first variable.Consider an optimal control system of the form Minimize J u(•) = ∞ a L x(t),u(t),t dt (1.1) over all admissible controls u(•) ∈ L 2 ([a,∞);U), where the trajectories x : [a,∞) → R n are differentiable on [a,∞) and satisfy the dynamic system ẋ(t) = f x(t),u(t),t , x(a) = x 0 . (1.2) From control theory, the well-known Pontryagin maximum principle, an important necessary optimality condition, is usually applied to get optimal controls for this system.By doing this, the following infinite-horizon Hamiltonian system is derived: Here, H(x, p,t) = λL(x, ū,t) + p, f (x, ū,t) is the Hamiltonian function for (1.1)-(1.2), •, • stands for inner product in R n , ū is an optimal control, and x(t) is the optimal trajectory corresponding to the optimal control ū.
The existence and uniqueness of solutions for system (1.3) is a very interesting question; if solutions to (1.3) are unique, then the optimal control for system (1.1)-(1.2) can be solved analytically or numerically through (1.3).When we consider the generalization of (1.3) in infinite-dimensional spaces, the following Hamiltonian system is obtained: where both x(t) and p(t) take values in a Hilbert space X for a ≤ t < ∞.It is always assumed that F,G : X × X × [a,∞) → X are nonlinear operators, that A(t) is a closed operator for each t ∈ [a,∞), and that A * (t) is the adjoint operator of A(t).
The following system is called a linear Hamiltonian system, which is a special case of (1.4), where ϕ(•),ψ(•) ∈ L 2 ([a,∞);X), and B(t), C(t) are selfadjoint linear operators from X to X for all t ∈ [a,∞).
In [2], Lions has discussed the existence and uniqueness of solutions for system (1.5) and gave an existence and uniqueness result.In [1], Hu and Peng considered the existence and uniqueness of solutions for a class of nonlinear forward-backward stochastic differential equations similar to (1.3) but on finite horizon, they provided an existence and uniqueness theorem for (1.3).Peng and Shi in [3] dealt with the existence and uniqueness of solutions for (1.3) using the techniques developed in [1].In this paper, we consider the existence and uniqueness of solutions for infinite-dimensional system (1.4).
Throughout the paper, the following basic assumptions hold.(I) There exists a real number L > 0 such that

Lemmas
Two lemmas are given in this section.They are essential to prove the main theorem.
Lemma 2.1.Consider the Hamiltonian system ) The functions F β and G β are defined as Assume that (2.1) has a unique solution for some real number β = β 0 ≥ 0 and any ϕ(t), ψ
Proof.For any given ϕ(•),ψ(•),x(•), p(•) ∈ L 2 ([a,∞);X) and δ > 0, construct the following Hamiltonian system: = αδx + δG(x, p,t). (2.4) The assumption of Lemma 2.1 implies that (2.3) has a unique solution for each pair given by .9) implies that implies that (2.12) Lianwen Wang 841 It follows from the estimates for I 1 , I 2 , and the assumption (I) that (2.13) Therefore, (2.14) Integrating between a and t k , we have (2.15) Letting k → ∞ and noting that (2.7), we obtain then J is a contractive mapping and hence has a unique fixed point.Thus, (2.3) becomes (3.4)This is a special case of system (1.5).Therefore, system (3.4) has a unique solution if both BR −1 B and Q are positive definite.