Discrete matrix Riccati equations with superposition formulas

An ordinary differential equation is said to have a superposition formula if its general solution can be expressed as a function of a finite number of particular solution. Nonlinear ODE's with superposition formulas include matrix Riccati equations. Here we shall describe discretizations of Riccati equations that preserve the superposition formulas. The approach is general enough to include $q$-derivatives and standard discrete derivatives.Comment: 20 pages; v.2: a misprint correcte

A Linear-Quadratic Optimal Control Problem for Mean-Field Stochastic Differential Equations in Infinite Horizon

A linear-quadratic (LQ, for short) optimal control problem is considered for mean-field stochastic differential equations with constant coefficients in an infinite horizon. The stabilizability of the control system is studied followed by the discussion of the well-posedness of the LQ problem. The optimal control can be expressed as a linear state feedback involving the state and its mean, through the solutions of two algebraic Riccati equations. The solvability of such kind of Riccati equations is investigated by means of semi-definite programming method.Comment: 40 page

An example of interplay between Physics and Mathematics: Exact resolution of a new class of Riccati Equations

A novel recipe for exactly solving in finite terms a class of special differential Riccati equations is reported. Our procedure is entirely based on a successful resolution strategy quite recently applied to quantum dynamical time-dependent SU(2) problems. The general integral of exemplary differential Riccati equations, not previously considered in the specialized literature, is explicitly determined to illustrate both mathematical usefulness and easiness of applicability of our proposed treatment. The possibility of exploiting the general integral of a given differential Riccati equation to solve an SU(2) quantum dynamical problem, is succinctly pointed out.Comment: 10 page