Differential Riccati equations for the Bolza problem associated with point boundary control of singular estimate control systems
Recent developments in boundary and point control theory have provided strong motivation for studying Riccati equations with unbounded coefficients. By unbounded coefficients we mean the coefficients of the Riccati equations that are given in terms of unbounded, and even uncloseable, operators. These typically result from unbounded control actions or unbounded observations. The unboundedness of the coefficients is, of course, a source of mathematical difficulties. Standard methods for establishing wellposedness to these nonlinear equations are no longer applicable. The problem is particularly acute when the coefficients in the nonlinear term of the Riccati equation are unbounded. This latter case corresponds to the unbounded control action. In these cases, the well-posedness of Riccati equations may fail altogether, as evidenced in . However, the situation is very different when the dynamics is generated by an analytic semigroup. This is to say that eAt is a generator of an analytic semigroup on a given Hilbert space H. For this class of dynamics the Riccati theory is by now well understood. The regularizing effect of analyticity compensates for the unboundedness of the coefficients, leading to a well-posed nonlinear problem.
Free and Moving Boundaries: Analysis, Simulation and Control
Lasiecka, I., & Tuffaha, A. (2007). Differential Riccati equations for the Bolza problem associated with point boundary control of singular estimate control systems. Free and Moving Boundaries: Analysis, Simulation and Control, 245-262. Retrieved from https://digitalcommons.memphis.edu/facpubs/4528