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Bounds on the solution of the time-varying linear matrix differential equation \(A'(t) = A^H(t) P(t) + P(t) A(t) + Q(t)\). (English) Zbl 1107.34031
Here, upper and lower bounds for the trace of the solution \(P(t)\) of the matrix differential equation \[ A'(t)=A^H(t)P(t)+P(t)A(t)+Q(t)\tag{\(*\)} \] are obtained, where \(A(t), Q(t)\in C^{n\times n}\) and \(A^H(t)\) denotes the complex conjugate transpose of \(A(t)\). These bounds are useful in a number of applications in system and control theory. A numerical example is given to verify these bounds.
The main theorem in the paper runs as follows: Consider (\(*\)) with \(P(t_0)=P_0=P_0^H\geq 0\). Let \(Q(t)=Q^H(t)\geq0\) and \(A(t)\) be continuous functions of \(t\). Then \[ \begin{aligned} \text{tr}(P(t))&\leq \text{tr}(P_0)e^{\int_{t_0}^t2\mu_M(A(\xi))\,d\xi}+\int_{t_0}^t \text{tr}(Q(\tau)) e^{\int_{\tau}^t2\mu_M(A(\xi))\,d\xi}\,d\tau,\\ \text{tr}(P(t))&\geq \text{tr}(P_0)e^{\int_{t_0}^t2\mu_m(A(\xi))\,d\xi}+ \int_{t_0}^t \text{tr}(Q(\tau))e^{\int_{\tau}^t2\mu_m(A(\xi))\,d\xi}\,d\tau, \end{aligned} \] for \(t\geq t_0\), where \(\text{tr}(P(t))\) denotes the trace of \(P(t)\), \[ \begin{aligned} \mu_M(A(t))&=\lambda_{\max}(\tfrac12(A(t)+A^H(t))),\\ \mu_m(A(t))&=\lambda_{\min}(\tfrac12(A(t)+A^H(t))), \end{aligned} \] \(\lambda_{\max}(A(t))\) and \(\lambda_{\min}(A(t))\) denote the maximum and minimum eigenvalues of \(A(t)\), respectively, and \(Q(t)>0\) denotes a positive semi-definite matrix.

MSC:
34C11 Growth and boundedness of solutions to ordinary differential equations
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