Necessary Optimality Conditions for Lyapunov-Type Optimization Problem with State Constraints

Abstract

This paper investigates a Lyapunov-type optimization problem incorporating an inequality integral constraint and a phase inequality constraint, both defined in L∞([0,T],Rn). The problem is approached directly using a theorem of the alternative, assuming differentiability, to derive necessary optimality conditions. Unlike earlier works, such as a nonsmooth convex isoperimetric problem with phase constraints or problems excluding phase constraints, this study focuses on a smooth case of problem. The problem formulation is motivated by the growing interest in continuous-time linear and nonlinear optimization problems, a field originally explored by Bellman (Proc Natl Acad Sci USA, 39:947–951, 1953). Building on prior results where new Karush–Kuhn–Tucker (KKT) conditions were introduced for problems with inequality constraints, this work extends the scope by incorporating an integral constraint into the analysis. The manuscript outlines the problem’s preliminaries, introduces a regularity condition for systems of inequalities, and presents a theorem of the alternative as the foundation for deriving the main results. Necessary optimality conditions are established and illustrated with an example, while the application of computational methods demonstrates how these conditions contribute to finding optimal solutions. By utilizing recent advancements in theorems of the alternative, this paper contributes to the theory of continuous-time programming problems in both finite-dimensional and infinite-dimensional spaces. The findings highlight the importance of integrating analytical and computational approaches to address these increasingly relevant optimization challenges