Glizer, Valery Y. Asymptotic solution of the singularly perturbed infinite dimensional Riccati equation. (English) Zbl 0916.49028 J. Math. Anal. Appl. 214, No. 1, 63-88 (1997). The main problem discussed by the author is the singularly perturbed linear-quadratic control posed in a Hilbert space setting. The author notes that if the corresponding operators are bounded then the problem can be modeled by Barbashin’s type integro-differential equations.He considers a linear system: \[ \begin{aligned} dx/dt & = A_1(t)x+ A_2(t)y+ B_1(t)u,\quad x(0)= x^0,\\ \varepsilon dy/dt & = A_3(t)x+ A_4(t)y+ B_2(t)u,\quad y(0)= y^0,\quad t\in[0, T].\end{aligned} \] Here, \(x\), \(y\) are state variables in Hilbert spaces \(H_x\) and \(H_y\), respectively, \(u\) is the control in the Hilbert space \(H_u\), and \(\varepsilon> 0\) is a “small” parameter. The reviewer is somewhat troubled by author’s identification of the tangent space with the integral curves \((A:H_{x\to}H_x)\), etc…, but hopefully no harm is done in the present discussion.In the same spirit, the author introduces the Hilbert space \(H_z\), which is the direct sum of mutually orthogonal spaces \(H_x\) and \(H_y\). The resulting equations are \(dz/dt= A(t,\varepsilon)z+ B(t,\varepsilon)u\), \(z(0)= z^0\). The cost functional is \[ J= (z(T), F(z(T)))+ \int^T_0 \{\langle z,D(t)z\rangle_{H_z}+ \langle u,Ru\rangle_{H_u}\}dt, \] where \(\langle,\rangle\) denotes an inner product, \(F\) is a positive definite and selfadjoint operator: \[ F=\begin{matrix} \lceil F_1, & F_2\rceil \\ \lfloor F^*_2, & F_3\rfloor\end{matrix}. \] Then \(u\) can be represented as \(u= -R^{-1} B^*K(t,\varepsilon)z\). The operator \(K\) satisfies the differential equation: \[ dK/dt= -KA(t,\varepsilon)- A^*(t,\varepsilon)K= KS(t,\varepsilon)K- D(t), \] and \(S(t,\varepsilon)= BR^{-1}B^*\). (Here, the author does not specify what kind of derivative of the operator \(K\) is considered, even if it is easy to guess.) To avoid singularities, the author assumes that the solutions are of the form: \[ K(t,\varepsilon)=\begin{matrix} \lceil K_1,& \varepsilon K_2\rceil \\ \lfloor\varepsilon K^*_2, & \varepsilon K_3\rfloor\end{matrix}. \] The equations for the operator \(K\) are transformed to an easier form of three equations in a new matrix variable, and asymptotic solutions are sought for small value of \(\varepsilon\). This result takes about twenty pages to derive. This is a quite involved process, and following some rather difficult calculations, the author restricts the asymptotic solution to the first-order. Under certain hypothesis, he proves existence of an optimal control. Reviewer: Vadim Komkov (Florida) Cited in 3 Documents MSC: 49N10 Linear-quadratic optimal control problems 34E15 Singular perturbations for ordinary differential equations 34G20 Nonlinear differential equations in abstract spaces 34H05 Control problems involving ordinary differential equations 49K40 Sensitivity, stability, well-posedness 93C15 Control/observation systems governed by ordinary differential equations Keywords:Riccati equations; singularly perturbed linear-quadratic control; Hilbert space; asymptotic solutions; existence of an optimal control PDFBibTeX XMLCite \textit{V. Y. Glizer}, J. Math. Anal. Appl. 214, No. 1, 63--88 (1997; Zbl 0916.49028) Full Text: DOI References: [1] Kalman, R. E., Contributions to the theory of optimal control, Bol. Soc. Mat. Mexicana, 5, 102-119 (1960) · Zbl 0112.06303 [2] Falb, P. L.; Kleinman, D. L., Remarks on the infinite dimensional Riccati equation, IEEE Trans. Automat. Control, 11, 534-536 (1966) [3] Lions, J. L., Optimal Control of Systems Governed by Partial Differential Equations (1971), Springer-Verlag: Springer-Verlag New York · Zbl 0203.09001 [4] Datko, R., A linear control problem in an abstract Hilbert space, J. Differential Equations, 9, 346-359 (1971) · Zbl 0218.93011 [5] Curtain, R. F.; Pritchard, A. J., The infinite-dimensional Riccati equation, J. Math. Anal. Appl., 47, 43-57 (1974) · Zbl 0279.93048 [6] Lasiecka, I.; Triggiani, R., Riccati differential equations with unbounded coefficients and nonsmooth terminal condition—The case of analytic semigroups, SIAM J. Math. Anal., 23, 449-481 (1992) · Zbl 0774.34047 [7] Avalos, G.; Lasiecka, I., Differential Riccati equation for the active control of a problem in structural acoustics, J. Optim. Theory Appl., 91, 695-728 (1996) · Zbl 0869.93023 [8] Kokotovic, P. V.; Yackel, R. A., Singular perturbation of linear regulators: Basic theorems, IEEE Trans. Automat. Control, 17, 29-37 (1972) · Zbl 0291.93023 [9] O’Malley, R. E.; Kung, C. F., On the matrix Riccati approach to a singularly perturbed regulator problem, J. Differential Equations, 16, 413-427 (1974) · Zbl 0289.34082 [10] Glizer, V. Y.; Dmitriev, M. G., Asymptotic properties of the solution of a singularly perturbed Cauchy problem encountered in optimal control theory, Differentsial’nye Uravneniya, 14, 601-612 (1978) · Zbl 0389.93021 [11] Daletskii, Yu. L.; Krein, M. G., Stability of Solutions of Differential Equations in Banach Space (1970), Nauka: Nauka Moscow [12] Langer, H., Über stark gedämpfte Scharen im Hilbertraum, J. Math. Mech., 17, 685-706 (1968) · Zbl 0157.21303 [13] Kühne, R., Minimaxprinzipe für stark gedämpfte Scharen, Acta Sci. Math. (Szeged), 29, 39-68 (1968) · Zbl 0169.17101 [14] Barbashin, E. A., Introduction to the Theory of Stability (1970), Wolters-Noordhoff: Wolters-Noordhoff Groningen · Zbl 0198.19703 [15] Glizer, V. Y.; Dmitriev, M. G., Asymptotic properties of a solution of singularly perturbed problem related to the penalty-function method, Differentsial’nye Uravneniya, 17, 1574-1580 (1981) [16] O’Malley, R. E.; Jameson, A., Singular perturbations and singular arcs, Part 1, IEEE Trans. Automat. Control, 20, 218-226 (1975) · Zbl 0298.49020 [17] Soule, J. L., Linear Operators in Hilbert Space (1968), Gordon & Breach: Gordon & Breach New York · Zbl 0182.45801 [18] Zabczyk, J., Remarks on the algebraic Riccati equation in Hilbert space, Appl. Math. Optim., 2, 251-258 (1976) [19] O’Malley, R. E., Singular Perturbation Methods for Ordinary Differential Equations (1991), Springer-Verlag: Springer-Verlag New York · Zbl 0743.34059 [20] Vasil’eva, A. B.; Butuzov, V. F.; Kalachev, L. V., The Boundary Function Method for Singular Perturbation Problems (1995), SIAM: SIAM Philadelphia · Zbl 0823.34059 [21] Trenogin, V. A., Asymptotic behavior and existence of a solution of the Cauchy problem for a first order differential equation with a small parameter in a Banach space, Dokl. Akad. Nauk, 152, 63-66 (1963) · Zbl 0143.11001 [23] Schwartz, L., Distributions a valeurs vectorielles, 1, Ann. Inst. Fourier, 7, 1-141 (1957) · Zbl 0089.09601 [24] Schwartz, L., Distributions a valeurs vectorielles, 2, Ann. Inst. Fourier, 8, 1-209 (1958) · Zbl 0089.09801 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.