Existence of Solutions of Riccati Differential Equations

Abstract

A necessary condition for the existence of the solution of the Riccati differential equation for both linear, time varying systems and nonlinear systems is introduced. First, a necessary condition for the existence of the solution of the Riccati differential equation for linear, time varying systems is proposed. Then, the sufficient conditions to satisfy the necessary condition are given. After that, the existence of the solution of the Riccati differential equation is generalized for nonlinear systems.

Tracking control of elastic joint parallel robots via state-dependent Riccati equation

“…are pointwise detectable and pointwise stabilizable, respectively [3,6,33]. If the state-dependent controllability matrix has full rank, the stabilizability condition is satisfied.…”

A max-plus primal space fundamental solution for a class of differential Riccati equations

“…Lemma A.1: Under Assumption 3.1, controllability of the dynamics (18) implies that Q 22 t ∈ S n×n >M for all t ∈ R >0 . Proof: (Lemma A.1) With M ∈ S n×n satisfying Assumption 3.1, recall that t * (M ) = +∞ as per (28).…”

“…see [5], [17], [18]. This maximal horizon of existence is either strictly positive and finite, or infinite.…”

“…(Necessity) Suppose that G t (x, y) ∈ R. Recalling the value function interpretation of G t (x, y), if the dynamics (18) are not controllable from x to y in time t, it immediately follows by definition (36) that G t (x, y) = −∞. Hence, the dynamics (18) must be controllable from x to y in time t. Necessity follows as x, y ∈ R n and t ∈ R >0 are arbitrary.…”

“…In developing a max-plus fundamental solution for DRE (1), (2) via Theorem 3.6, it is useful to establish a connection between finiteness of the kernel G t of (36) and controllability of the underlying dynamics (18). Assumption 3.8: (A, B) of (18) is controllable.…”