Pseudo-Riccati operator equations and feedback optimal control of semilinear systems

Concerns the quadratic optimal control of semilinear evolutionary systems. It is known that if a viscosity solution of the associated Hamilton-Jacobi equation exists, then its generalized derivative will provide a feedback optimal control. However, the global existence of a viscosity solution of the infinite-dimensional Hamilton-Jacobi equations remains open. Moreover, a real-time synthesis via a set-valued selection due to an involved generalized derivative is practically not implementable. A synthesis approach is presented for the semilinear evolution equations with a separate linear control term and a quadratic criterion of optimisation during a finite time interval. This approach features the pseudo-Riccati equation whose solution directly provides a feedback operator of the closed-loop optimal control.