Robust Optimal Switching Control for Nonlinear Systems

We formulate a robust optimal control problem for a general nonlinear system with finitely many admissible control settings and with costs assigned to switching of controls. We formulate the problem both in an L2-gain/dissipative system framework and in a game-theoretic framework. We show that, under appropriate assumptions, a continuous switching-storage function is characterized as a viscosity supersolution of the appropriate system of quasivariational inequalities (the appropriate generalization of the Hamilton-Jacobi-Bellman-Isaacs equation for this context), and that the minimal such switching-storage function is equal to the continuous switching lower-value function for the game. Finally we show how a prototypical example with one-dimensional state space can be solved by a direct geometric construction.