Dynamic Value functions and optimal control in the presence of input saturation

The optimal control of nonlinear input-affine systems is tackled in the presence of input saturation. The problem is formulated within the framework of the Dynamic Programming approach, which hinges upon the solution of the Hamilton-Jacobi-Bellman partial differential equation. The notion of Dynamic Value function is extended herein to inputaffine nonlinear systems in the presence of input saturation, providing, in general, a time-varying dynamic control law that approximates the solution of the optimal control problem with bounded input. Then, the second part of the paper discusses a systematic procedure to construct a Dynamic Value function without requiring the solution of any partial differential equation or inequality. This construction relies upon the notion of algebraic solution of the Hamilton-Jacobi-Bellman partial differential equation.