Boundary Value Problem for an Operator-Differential Riccati Equation in the Hilbert Space on the Interval

ABSTRACT

The paper is devoted to obtaining the necessary and sufficient conditions of the solvability of weakly perturbed boundary-value problems for the nonlinear operator-differential Riccati equation in the Hilbert space on the interval and whole line with parameter ?. We find the solution of the given boundary value problem which for ε = 0 turns in one of the solutions of generating boundary value problem. Solution of the generating problem is constructed with the using generalized operator in analytical form. Iterative process for finding of solutions of weakly nonlinear equation with quadratic error is constructed.

The paper is devoted to obtaining the necessary and sufficient conditions of the solvability of weakly perturbed boundary-value problems for the nonlinear operator-differential Riccati equation in the Hilbert space on the interval and whole line with parameter ?. We find the solution of the given boundary value problem which for ε = 0 turns in one of the solutions of generating boundary value problem. Solution of the generating problem is constructed with the using generalized operator in analytical form. Iterative process for finding of solutions of weakly nonlinear equation with quadratic error is constructed.

Cite this paper

Pokutnyi, O. (2015) Boundary Value Problem for an Operator-Differential Riccati Equation in the Hilbert Space on the Interval.*Advances in Pure Mathematics*, **5**, 865-873. doi: 10.4236/apm.2015.514081.

Pokutnyi, O. (2015) Boundary Value Problem for an Operator-Differential Riccati Equation in the Hilbert Space on the Interval.

References

[1] Wyss, C. (2008) Perturbation Theory for Hamiltonian Operator Matrices and Riccati Equations. Dissertation, Bern, 164 p.

[2] Abbasbandy, S. (2006) Iterated He’s Homotopy Perturbation Method for Quadratic Riccati Differential Equation. Applied Mathematics and Computation, 175, 581-589.

http://dx.doi.org/10.1016/j.amc.2005.07.035

[3] Sragovich, V.G. and Chornik, (1995) About Asymptotical Behavior of Solutions of Algebraic Riccati Equation for Continuous Time. Automation and Remote Control, No. 8, 90-92.

[4] Lazareva, A.B. and Pakshin, P.V. (1986) Solution of Matrix Lurier, Riccati and Lyapunov Equations for Digital Systems. Automation and Remote Control, No. 12, 17-22.

[5] Chu, E.K.-W., Fan, H.-Y. and Lin, W.-W. (2005) A Structure-Preserving Doubling Algorithm for Continuous-Time Algebraic Riccati Equations. Linear Algebra and Its Applications, 396, 55-80.

http://dx.doi.org/10.1016/j.laa.2004.10.010

[6] Freiling, G. (2002) A Survey of Nonsymmetric Riccati Equations. Linear Algebra and Its Applications, 351-352, 243-270.

http://dx.doi.org/10.1016/S0024-3795(01)00534-1

[7] Hiramoto, K., Doki, H. and Obinata, G. (2000) Optimal Sensor/Actuator Placement for Active Vibration Control Using Explicit Solution of Algebraic Riccati Equation. Journal of Sound and Vibration, 229, 1057-1075.

http://dx.doi.org/10.1006/jsvi.1999.2530

[8] Pron’kin, V.S. (1994) On Quasiperiodic Solutions of the Matrix Riccati Equation. Russian Academy of Sciences, Izvestiya Mathematics, 43, 455-470.

http://dx.doi.org/10.1070/im1994v043n03abeh001575

[9] Osmolovskii, N.P. and Lempio, F. (2000) Jacobi Conditions and the Riccati Equation for a Broken Extremal. Journal of Mathematical Sciences, 100, 2572-2592.

http://dx.doi.org/10.1007/BF02673843

[10] Grigoryan, G.A. (2015) Global Solvability of Scalar Riccati Equations. Russian Mathematics (Izvestiya VUZ. Matematika), 59, 31-42.

[11] Boichuk, O.A. and Krivosheya, S.A. (1998) Criterion of the Solvability of Matrix Lyapunov Type Equations. Ukrainian Mathematical Journal, 50, 1021-1026.

http://dx.doi.org/10.1007/BF02513089

[12] Boichuk, A.A. and Krivosheya, S.A. (2001) A Critical Periodic Boundary-Value Problem for a Matrix Riccati Equation. Differential Equations, 37, 464-471.

http://dx.doi.org/10.1023/A:1019267220924

[13] Krein, M.G. (1970) Stability of Solutions of Differential Equations in Banach Space. Science, Moscow, 534 p. (In Russian)

[14] Boichuk, A.A. and Samoilenko, A.M. (2004) Generalized Inverse Operators and Fredholm Boundary-Value Problems. VSP, Utrecht-Boston, 317 p.

http://dx.doi.org/10.1515/9783110944679

[15] Pokutnyi, O.O. (2013) Generalized-Invertible Operator in the Banach Hilbert and Frechet Spaces. Visnik of the Kiev National Taras Shevchenko University, Series of Physical and Mathematical Sciences.

[16] Panasenko, E.V. and Pokutnyi, O.O. (2013) Boundary Value Problems for Differential Equations in Banach Space with Unbounded Operator in Linear Part. Nonlinear Oscillations, 16, 518-526.

[17] Pokutnyi, A.A. (2012) Bounded Solutions of Linear and Weakly Nonlinear Differential Equations in Banach Space with Unbounded Linear Part. Differential Equations, 48, 803-813.

http://dx.doi.org/10.1134/S0012266112060055

[1] Wyss, C. (2008) Perturbation Theory for Hamiltonian Operator Matrices and Riccati Equations. Dissertation, Bern, 164 p.

[2] Abbasbandy, S. (2006) Iterated He’s Homotopy Perturbation Method for Quadratic Riccati Differential Equation. Applied Mathematics and Computation, 175, 581-589.

http://dx.doi.org/10.1016/j.amc.2005.07.035

[3] Sragovich, V.G. and Chornik, (1995) About Asymptotical Behavior of Solutions of Algebraic Riccati Equation for Continuous Time. Automation and Remote Control, No. 8, 90-92.

[4] Lazareva, A.B. and Pakshin, P.V. (1986) Solution of Matrix Lurier, Riccati and Lyapunov Equations for Digital Systems. Automation and Remote Control, No. 12, 17-22.

[5] Chu, E.K.-W., Fan, H.-Y. and Lin, W.-W. (2005) A Structure-Preserving Doubling Algorithm for Continuous-Time Algebraic Riccati Equations. Linear Algebra and Its Applications, 396, 55-80.

http://dx.doi.org/10.1016/j.laa.2004.10.010

[6] Freiling, G. (2002) A Survey of Nonsymmetric Riccati Equations. Linear Algebra and Its Applications, 351-352, 243-270.

http://dx.doi.org/10.1016/S0024-3795(01)00534-1

[7] Hiramoto, K., Doki, H. and Obinata, G. (2000) Optimal Sensor/Actuator Placement for Active Vibration Control Using Explicit Solution of Algebraic Riccati Equation. Journal of Sound and Vibration, 229, 1057-1075.

http://dx.doi.org/10.1006/jsvi.1999.2530

[8] Pron’kin, V.S. (1994) On Quasiperiodic Solutions of the Matrix Riccati Equation. Russian Academy of Sciences, Izvestiya Mathematics, 43, 455-470.

http://dx.doi.org/10.1070/im1994v043n03abeh001575

[9] Osmolovskii, N.P. and Lempio, F. (2000) Jacobi Conditions and the Riccati Equation for a Broken Extremal. Journal of Mathematical Sciences, 100, 2572-2592.

http://dx.doi.org/10.1007/BF02673843

[10] Grigoryan, G.A. (2015) Global Solvability of Scalar Riccati Equations. Russian Mathematics (Izvestiya VUZ. Matematika), 59, 31-42.

[11] Boichuk, O.A. and Krivosheya, S.A. (1998) Criterion of the Solvability of Matrix Lyapunov Type Equations. Ukrainian Mathematical Journal, 50, 1021-1026.

http://dx.doi.org/10.1007/BF02513089

[12] Boichuk, A.A. and Krivosheya, S.A. (2001) A Critical Periodic Boundary-Value Problem for a Matrix Riccati Equation. Differential Equations, 37, 464-471.

http://dx.doi.org/10.1023/A:1019267220924

[13] Krein, M.G. (1970) Stability of Solutions of Differential Equations in Banach Space. Science, Moscow, 534 p. (In Russian)

[14] Boichuk, A.A. and Samoilenko, A.M. (2004) Generalized Inverse Operators and Fredholm Boundary-Value Problems. VSP, Utrecht-Boston, 317 p.

http://dx.doi.org/10.1515/9783110944679

[15] Pokutnyi, O.O. (2013) Generalized-Invertible Operator in the Banach Hilbert and Frechet Spaces. Visnik of the Kiev National Taras Shevchenko University, Series of Physical and Mathematical Sciences.

[16] Panasenko, E.V. and Pokutnyi, O.O. (2013) Boundary Value Problems for Differential Equations in Banach Space with Unbounded Operator in Linear Part. Nonlinear Oscillations, 16, 518-526.

[17] Pokutnyi, A.A. (2012) Bounded Solutions of Linear and Weakly Nonlinear Differential Equations in Banach Space with Unbounded Linear Part. Differential Equations, 48, 803-813.

http://dx.doi.org/10.1134/S0012266112060055