Optimal control using an algebraic method for control-affine non-linear systems

A deterministic optimal control problem is solved for a control-affine nonlin-ear system with a nonquadratic cost function. We algebraically solve the Hamilton-Jacobi equation for the gradient of the value function. This eliminates the need to explicitly solve the solution of a Hamilton-Jacobi partial differential equation. We interpret the value function in terms of the control Lyapunov function. Then we provide the stabilizing controller and the stability margins. Furthermore, we derive an optimal controller for a control-affine nonlinear system using the state dependent Riccati equation (SDRE) method; this method gives a similar optimal controller as the controller from the algebraic method. We also find the optimal controller when the cost function is the exponential-of-integral case, which is known as risk-sensitive (RS) control. Finally, we show that SDRE and RS methods give equivalent optimal controllers for nonlinear deterministic systems. Examples demonstrate the proposed methods.