Abstract
In this paper, we are concerned with a class of optimal control problem governed by nonlinear first order dynamic equation on time scales. By imposing some suitable conditions on the related functions, for any given control policy, we first obtain the existence of a unique solution for the nonlinear controlled system. Then, we study the existence of an optimal solution for the optimal control problem.
Highlights
Suppose that there is a flock of sheep in a pasture
By imposing some suitable conditions on p, f, and g, for any given control policy u ∈ Uad, we obtain the existence of a unique solution xu for the nonlinear controlled system (5). en, we study the optimal control problem (P)
We always suppose that the control space is C([0, T]T, R) and the admissible control set Uad is a compact subset of C([0, T]T, R)
Summary
Suppose that there is a flock of sheep in a pasture. We consider the changes in the number of sheep during a time interval [0, σ(T)]T. En, for any given control policy u ∈ Uad, it is easy to know that the changes in the number of sheep can be described by the following linear dynamic equation: xΔ(t) + p(t)x(σ(t)) r(t) + q(t)u(t), t ∈ [0, T]T . Motivated greatly by the abovementioned works, in this paper, we suppose that the controlled system is governed by the following more general nonlinear periodic boundary value problem:
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