Abstract
We discuss the existence of optimal controls for a Lagrange problem of systems governed by the second-order nonlinear impulsive differential equations in infinite dimensional spaces. We apply a direct approach to derive the maximum principle for the problem at hand. An example is also presented to demonstrate the theory.
Highlights
It is well known that Pontryagin maximum principle plays a central role in optimal control theory
In 1960, Pontryagin derived the maximum principle for optimal control problems in finite dimensional spaces
There are a few papers addressing the existence of optimal controls for the systems governed by the second-order nonlinear impulsive differential equations
Summary
It is well known that Pontryagin maximum principle plays a central role in optimal control theory. In 1960, Pontryagin derived the maximum principle for optimal control problems in finite dimensional spaces (see [1]). The maximum principle for optimal control problems involving first-order nonlinear impulsive differential equations in finite (or infinite) dimensional spaces has been extensively studied (see [2–10]). There are a few papers addressing the existence of optimal controls for the systems governed by the second-order nonlinear impulsive differential equations. Peng and Xiang [11, 12] applied the semigroup theory to establish the existence of optimal controls for a class of second-order nonlinear differential equations in infinite dimensional spaces. We consider a Lagrange problem of system governed by (1.1) and prove the existence of optimal controls.
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