Existence and Optimal Control Results for Second-Order Semilinear System in Hilbert Spaces

Abstract

The article provides sufficient conditions for the existence of optimal control for second-order semilinear control system in Hilbert spaces. We consider the integral cost function as $$\begin{aligned} J(z,v):=\int _{0}^{T}L(\tau ,z^{v}(\tau ),v(\tau ))\mathrm{d}t, \end{aligned}$$ subject to the equations $$\begin{aligned} z''(\tau )= & {} Az(\tau )+Bv(\tau )+g(\tau ,z(\tau ));\,0<\tau \le T\\ z(0)= & {} z_0\\ z'(0)= & {} z_1. \end{aligned}$$ Next, we discuss the existence and the uniqueness of mild solutions for the above proposed problem using Banach fixed point theorem. The stated Lagrange’s problem admits at least one optimal control pair under certain assumptions. Finally, the validation of theoretical results is provided through an example.