Analytical infinite-time optimal and sub-optimal controllers for scalar nonlinear systems using krotov sufficient conditions

This work considers the problem of obtaining analytical expressions of optimal control laws for scalar nonlinear systems. For a nonlinear system, usually the optimal control laws are computed using iterative/sequential methods. To circumvent this issue, Krotov sufficient conditions, are employed to compute analytical optimal control laws. These conditions work upon the idea of translating the optimal control problem to an optimization problem by using the so-called extension principle and an ad-hoc selection of a function called Krotov function. Subsequently, an iterative solution of this optimization problem is computed. The angle of our attack is modify these equivalent optimization problems via suitable selection of Krotov function so as to obtain a non-iterative solution. The sufficient conditions for the existence of analytical expressions for optimal control laws for a scalar nonlinear system (affine in the control input) with a quadratic cost are derived. The proposed method is illustrated by numerical examples.