Piecewise linear quadratic optimal control

The use of piecewise quadratic cost functions is extended from stability analysis of piecewise linear systems to performance analysis and optimal control. Lower bounds on the optimal control cost are obtained by semi-definite programming based on the Bellman inequality. This also gives an approximation to the optimal control law. An upper bound to the optimal cost is obtained by another convex optimization problem using the given control law. A compact matrix notation is intoduced to support the calculations and it is proved that the framework of piecewise linear systems can be used to analyse smooth nonlinear dynamics with arbitrary accuracy. Keywords—Nonlinear systems, optimal control, semi-definite programming