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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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