Log–log convexity of an optimal control problem for positive linear systems

Abstract

This paper investigates the finite-time optimal control problems for positive linear systems with a time-varying control input. Cost functional of the system state is constructed. Under some assumptions on the designed parameters and cost functionals, the optimization problem with piecewise-constant matrix functions is first proved to be log–log convex and can be solved via geometric programming. Then, the log–log convex result is further extended to the optimization problem with continuous functions. Finally, an optimal control problem is investigated to verify the effectiveness of the proposed optimization framework.