Integrability conditions for Lie systems are related to reduction or transformation processes. We here analyse a geometric method construct integrability Riccati equations following these approaches. This approach provides us with unified geometrical viewpoint that allows some previous works on the topic and explain new properties. Moreover, this can be straightforwardly generalised describe an...

We discuss a quadratic criterion optimal control problem for stochastic linear system with delay in both state and control variables. This problem will lead to a kind of generalized forward-backward stochastic differential equations FBSDEs with Itô’s stochastic delay equations as forward equations and anticipated backward stochastic differential equations as backward equations. Especially, we p...

By using the generalized Riccati transformation and the inequality technique, we establish a oscillation criterion for certain non-linear secondorder dynamic equations with damping on a time scale.

Several instances of integrable Riccati equations are analyzed from the geometric perspective of the theory of Lie systems. This provides us a unifying viewpoint for previous approaches.

For timed event graphs, linear models were obtained using dioid algebra. After describing backward equations which solve an optimal tracking problem and which introduce co-state variables, this paper presents preliminary results concerning the matrix of ‘ratios’ (i.e. conventional differences) of co-states over states: this matrix sounds like a Riccati matrix, although a neat analogue to a Ricc...

The paper describes the set of solutions of the discrete-time algebraic Riccati equation. It is shown that each solution is a combination of a pair of opposite unmixed solutions. There is a one-to-one correspondence between solutions and invariant subspaces of the closed loop matrix of an unmixed solution. The results of the paper provide an extended counterpart of the parametrization theory of...

We consider the numerical solution of the rational algebraic Riccati equations in Rn, arising from stochastic optimal control in continuousand discrete-time. Applying the homotopy method, we continue from the maximal stabilizing solutions of the deterministic algebraic Riccati equations, which are readily available. The associated differential equations require the solutions of some generalized...

This article concerns the oscillation of second-order nonlinear dynamic equations. By using generalized Riccati transformations, Kiguradzetype and Belohorec-type oscillation theorems are obtained on an arbitrary time scale. Our results cover those for differential equations and difference equations, and provide new oscillation criteria for irregular time scales. Some examples are given to illus...

We consider passivity-preserving model reduction of circuit equations using the bounded real balanced truncation method applied to a Moebius-transformed system. This method is based on balancing the solutions of the projected Lur’e or Riccati matrix equations. We also discuss their numerical solution exploiting the underlying structure of circuit equations. A numerical example is given.