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Phase portrait of the matrix Riccati equation. (English) Zbl 0594.34044

The matrix Riccati equations \(dK/dt=B_{21}+B_{22}K-KB_{11}- KB_{12}K\) (where \(K=K(t)\in {\mathbb{R}}^{m\times n}\) is variable and \(B_{12}\in {\mathbb{R}}^{m\times n}\), \(B_{22}\in {\mathbb{R}}^{m\times m}\), \(B_{11}\in {\mathbb{R}}^{n\times n}\), \(B_{12}\in {\mathbb{R}}^{n\times m}\) are constant matrices) and \(dK/dt=-Q-A'K-KA+KLK\) (where \(K=K(t)\in {\mathbb{R}}^{n\times n}\) is symmetric and \(Q,A,L\in {\mathbb{R}}^{n\times n}\) are constant, L and Q symmetric, L nonnegative definite) are thoroughly analysed under some semisimplicity-like assumptions.
The original equations are extended on the compactifications of the underlying spaces \({\mathbb{R}}^{n\times m}\) and \({\mathbb{R}}^{n\times n}\), to the Grassmann manifold \(G^ n({\mathbb{R}}^{n+m})\) of all n-dimensional linear subspaces of \({\mathbb{R}}^{n+m}\) and to the Lagrange-Grassmann manifold L(n) of all Lagrange subspaces of \({\mathbb{R}}^{2n}\) to a certain skew-symmetric bilinear form, respectively. Then the arising flows are induced by global actions of 1-parameter subgroups of \(Gl(n+m,{\mathbb{R}})\) and Sp(n,\({\mathbb{R}})\) so that the phase portraits are simplified.
We can mention only a few results. The nonwandering sets are unions of invariant torus, stable and unstable submanifolds of each tori are described by a Schubert cell decomposition. Although the vector fields are not of the Morse-Smale type, a version of Morse inequalities (in fact, equalities) is valid which permits one to determine the mod 2 Betti numbers of \(G^ n(n+m,{\mathbb{R}})\) and L(n). Analysing the neighbourhoods of the points at infinity, asymptotic behaviour of solutions can be derived. The paper includes a wide variety of other very concrete results and is of fundamental importance for specialists in dynamical systems and optimal control theory.
Reviewer: J.Chrastina

MSC:

34C30 Manifolds of solutions of ODE (MSC2000)
37C10 Dynamics induced by flows and semiflows
22E99 Lie groups
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