Upper and lower bounds for the solution of the general matrix Riccati differential equation on a time scale
The authors study the general matrix Riccati differential equations (GMRDE) on a time scale of the form \[ R^{\Delta}(t) = A(t) + B(t)R(t) + R(\Gamma (t)) B^{*} - R(\Gamma(t))C(t)R(t), \tag{1} \] where \(A(t)\) and \(C(t)\) are symmetric \(n\times n\)-matrices while \(B(t), V(t), T(t)\) and \(R(t)\) are \(n\times n\)-matrices, and the star denotes the transpose of the matrix. Some properties of time scale and operations on time scale are defined. Applying the quazilinearization technique to the general matrix Riccati differential equation on \(T\), the authors obtain the linear equation \[ U^{\Delta}(t)=[B(t)-T(t)C(t)U(t)+U(\gamma(t))][B^{*}(t)-C(t)T(t)]+ A(t)+ T(t)C(t)T(t) \tag{2} \] with \(U(0)=E\). The solution to the linearized equation in terms of fundamental matrices and an upper bound for the solution to equation (1) on \(T\) are obtained. In the next section, a lower bound for the solution to equation (1) is found. Finally, the authors study the monotonicity of successive approximations.
scientific article; zbMATH DE number 1758474
scientific article