The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
Peter Benner - One of the best experts on this subject based on the ideXlab platform.
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galerkin trial spaces and davison maki methods for the numerical solution of differential Riccati Equations
Applied Mathematics and Computation, 2021Co-Authors: Maximilian Behr, Peter Benner, Jan HeilandAbstract:Abstract The differential Riccati equation appears in different fields of applied mathematics like control and system theory. Recently, Galerkin methods based on Krylov subspaces were developed for the autonomous differential Riccati equation. These methods overcome the prohibitively large storage requirements and computational costs of the numerical solution. Known solution formulas are reviewed and extended. Because of memory-efficient approximations, invariant subspaces for a possibly low-dimensional solution representation are identified. A Galerkin projection onto a trial space related to a low-rank approximation of the solution of the algebraic Riccati equation is proposed. The modified Davison-Maki method is used for time discretization. Known stability issues of the Davison-Maki method are discussed. Numerical experiments for large-scale autonomous differential Riccati Equations and a comparison with high-order splitting schemes are presented.
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efficient solution of large scale algebraic Riccati Equations associated with index 2 daes via the inexact low rank newton adi method
Applied Numerical Mathematics, 2020Co-Authors: Peter Benner, Matthias Heinkenschloss, Jens Saak, Heiko K WeicheltAbstract:Abstract This paper extends the algorithm of Benner et al. (2016) [10] to Riccati Equations associated with Hessenberg index-2 Differential Algebratic Equation (DAE) systems. Such DAE systems arise, e.g., from semi-discretized, linearized (around steady state) Navier-Stokes Equations. The solution of the associated Riccati equation is important, e.g., to compute feedback laws that stabilize the Navier-Stokes Equations. Challenges in the numerical solution of the Riccati equation arise from the large-scale of the underlying systems and the algebraic constraint in the DAE system. These challenges are met by a careful extension of the inexact low-rank Newton-ADI method to the case of DAE systems. A main ingredient in the extension to the DAE case is the projection onto the manifold described by the algebraic constraints. In the algorithm, the Equations are never explicitly projected, but the projection is only applied as needed. Numerical experience indicates that the algorithmic choices for the control of inexactness and line-search can help avoid subproblems with matrices that are only marginally stable. The performance of the algorithm is illustrated on a large-scale Riccati equation associated with the stabilization of Navier-Stokes flow around a cylinder.
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a numerical comparison of different solvers for large scale continuous time algebraic Riccati Equations and lqr problems
SIAM Journal on Scientific Computing, 2020Co-Authors: Peter Benner, Patrick Kurschner, Zvonimir Bujanovic, Jens SaakAbstract:In this paper, we discuss numerical methods for solving large-scale continuous-time algebraic Riccati Equations. These methods have been the focus of intensive research in recent years, and signifi...
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invariant galerkin ansatz spaces and davison maki methods for the numerical solution of differential Riccati Equations
arXiv: Numerical Analysis, 2019Co-Authors: Maximilian Behr, Peter Benner, Jan HeilandAbstract:The differential Riccati equation appears in different fields of applied mathematics like control and system theory. Recently Galerkin methods based on Krylov subspaces were developed for the autonomous differential Riccati equation. These methods overcome the prohibitively large storage requirements and computational costs of the numerical solution. In view of memory efficient approximation, we review and extend known solution formulas and identify invariant subspaces for a possibly low-dimensional solution representation. Based on these theoretical findings, we propose a Galerkin projection onto a space related to a low-rank approximation of the algebraic Riccati equation. For the numerical implementation, we provide an alternative interpretation of the modified \emph{Davison-Maki method} via the transformed flow of the differential Riccati equation, which enables us to rule out known stability issues of the method in combination with the proposed projection scheme. We present numerical experiments for large-scale autonomous differential Riccati Equations and compare our approach with high-order splitting schemes.
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a numerical comparison of solvers for large scale continuous time algebraic Riccati Equations and lqr problems
arXiv: Numerical Analysis, 2018Co-Authors: Peter Benner, Patrick Kurschner, Zvonimir Bujanovic, Jens SaakAbstract:In this paper, we discuss numerical methods for solving large-scale continuous-time algebraic Riccati Equations. These methods have been the focus of intensive research in recent years, and significant progress has been made in both the theoretical understanding and efficient implementation of various competing algorithms. There are several goals of this manuscript: first, to gather in one place an overview of different approaches for solving large-scale Riccati Equations, and to point to the recent advances in each of them. Second, to analyze and compare the main computational ingredients of these algorithms, to detect their strong points and their potential bottlenecks. And finally, to compare the effective implementations of all methods on a set of relevant benchmark examples, giving an indication of their relative performance.
Jens Saak - One of the best experts on this subject based on the ideXlab platform.
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efficient solution of large scale algebraic Riccati Equations associated with index 2 daes via the inexact low rank newton adi method
Applied Numerical Mathematics, 2020Co-Authors: Peter Benner, Matthias Heinkenschloss, Jens Saak, Heiko K WeicheltAbstract:Abstract This paper extends the algorithm of Benner et al. (2016) [10] to Riccati Equations associated with Hessenberg index-2 Differential Algebratic Equation (DAE) systems. Such DAE systems arise, e.g., from semi-discretized, linearized (around steady state) Navier-Stokes Equations. The solution of the associated Riccati equation is important, e.g., to compute feedback laws that stabilize the Navier-Stokes Equations. Challenges in the numerical solution of the Riccati equation arise from the large-scale of the underlying systems and the algebraic constraint in the DAE system. These challenges are met by a careful extension of the inexact low-rank Newton-ADI method to the case of DAE systems. A main ingredient in the extension to the DAE case is the projection onto the manifold described by the algebraic constraints. In the algorithm, the Equations are never explicitly projected, but the projection is only applied as needed. Numerical experience indicates that the algorithmic choices for the control of inexactness and line-search can help avoid subproblems with matrices that are only marginally stable. The performance of the algorithm is illustrated on a large-scale Riccati equation associated with the stabilization of Navier-Stokes flow around a cylinder.
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a numerical comparison of different solvers for large scale continuous time algebraic Riccati Equations and lqr problems
SIAM Journal on Scientific Computing, 2020Co-Authors: Peter Benner, Patrick Kurschner, Zvonimir Bujanovic, Jens SaakAbstract:In this paper, we discuss numerical methods for solving large-scale continuous-time algebraic Riccati Equations. These methods have been the focus of intensive research in recent years, and signifi...
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a numerical comparison of solvers for large scale continuous time algebraic Riccati Equations and lqr problems
arXiv: Numerical Analysis, 2018Co-Authors: Peter Benner, Patrick Kurschner, Zvonimir Bujanovic, Jens SaakAbstract:In this paper, we discuss numerical methods for solving large-scale continuous-time algebraic Riccati Equations. These methods have been the focus of intensive research in recent years, and significant progress has been made in both the theoretical understanding and efficient implementation of various competing algorithms. There are several goals of this manuscript: first, to gather in one place an overview of different approaches for solving large-scale Riccati Equations, and to point to the recent advances in each of them. Second, to analyze and compare the main computational ingredients of these algorithms, to detect their strong points and their potential bottlenecks. And finally, to compare the effective implementations of all methods on a set of relevant benchmark examples, giving an indication of their relative performance.
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efficient solution of large scale algebraic Riccati Equations associated with index 2 daes via the inexact low rank newton adi method
arXiv: Numerical Analysis, 2018Co-Authors: Peter Benner, Matthias Heinkenschloss, Jens Saak, Heiko K WeicheltAbstract:This paper extends the algorithm of Benner, Heinkenschloss, Saak, and Weichelt: An inexact low-rank Newton-ADI method for large-scale algebraic Riccati Equations, Applied Numerical Mathematics Vol. 108 (2016), pp. 125-142, this https URL to Riccati Equations associated with Hessenberg index-2 Differential Algebraic Equation (DAE) systems. Such DAE systems arise, e.g., from semi-discretized, linearized (around steady state) Navier-Stokes Equations. The solution of the associated Riccati equation is important, e.g., to compute feedback laws that stabilize the Navier-Stokes Equations. Challenges in the numerical solution of the Riccati equation arise from the large-scale of the underlying systems, the algebraic constraint in the DAE system, and the fact that matrices arising in some subproblems may only be marginally stable. These challenges are met by a careful extension of the inexact low-rank Newton-ADI method to the case of DAE systems. A main ingredient in the extension to the DAE case is the projection onto the manifold of the algebraic constraints. In the algorithm, the Equations are never explicitly projected, but the projection is only applied as needed. The performance of the algorithm is illustrated on a large-scale Riccati equation associated with the stabilization of Navier-Stokes flow around a cylinder.
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RADI: a low-rank ADI-type algorithm for large scale algebraic Riccati Equations
Numerische Mathematik, 2018Co-Authors: Peter Benner, Patrick Kurschner, Zvonimir Bujanović, Jens SaakAbstract:This paper introduces a new algorithm for solving large-scale continuous-time algebraic Riccati Equations (CARE). The advantage of the new algorithm is in its immediate and efficient low-rank formulation, which is a generalization of the Cholesky-factored variant of the Lyapunov ADI method. We discuss important implementation aspects of the algorithm, such as reducing the use of complex arithmetic and shift selection strategies. We show that there is a very tight relation between the new algorithm and three other algorithms for CARE previously known in the literature—all of these seemingly different methods in fact produce exactly the same iterates when used with the same parameters: they are algorithmically different descriptions of the same approximation sequence to the Riccati solution.
Tony Stillfjord - One of the best experts on this subject based on the ideXlab platform.
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singular value decay of operator valued differential lyapunov and Riccati Equations
Siam Journal on Control and Optimization, 2018Co-Authors: Tony StillfjordAbstract:We consider operator-valued differential Lyapunov and Riccati Equations, where the operators $B$ and $C$ may be relatively unbounded with respect to $A$ (in the standard notation). In this setting,...
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Adaptive high-order splitting schemes for large-scale differential Riccati Equations
Numerical Algorithms, 2018Co-Authors: Tony StillfjordAbstract:We consider high-order splitting schemes for large-scale differential Riccati Equations. Such Equations arise in many different areas and are especially important within the field of optimal control. In the large-scale case, it is critical to employ structural properties of the matrix-valued solution, or the computational cost and storage requirements become infeasible. Our main contribution is therefore to formulate these high-order splitting schemes in an efficient way by utilizing a low-rank factorization. Previous results indicated that this was impossible for methods of order higher than 2, but our new approach overcomes these difficulties. In addition, we demonstrate that the proposed methods contain natural embedded error estimates. These may be used, e.g., for time step adaptivity, and our numerical experiments in this direction show promising results.
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singular value decay of operator valued differential lyapunov and Riccati Equations
arXiv: Optimization and Control, 2018Co-Authors: Tony StillfjordAbstract:We consider operator-valued differential Lyapunov and Riccati Equations, where the operators $B$ and $C$ may be relatively unbounded with respect to $A$ (in the standard notation). In this setting, we prove that the singular values of the solutions decay fast under certain conditions. In fact, the decay is exponential in the negative square root if $A$ generates an analytic semigroup and the range of $C$ has finite dimension. This extends previous similar results for algebraic Equations to the differential case. When the initial condition is zero, we also show that the singular values converge to zero as time goes to zero, with a certain rate that depends on the degree of unboundedness of $C$. A fast decay of the singular values corresponds to a low numerical rank, which is a critical feature in large-scale applications. The results reported here provide a theoretical foundation for the observation that, in practice, a low-rank factorization usually exists.
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low rank second order splitting of large scale differential Riccati Equations
IEEE Transactions on Automatic Control, 2015Co-Authors: Tony StillfjordAbstract:We apply first- and second-order splitting schemes to the differential Riccati equation. Such Equations are very important in, e.g., linear quadratic regulator (LQR) problems, where they provide a link between the state of the system and the optimal input. The methods can also be extended to generalized Riccati Equations, e.g., arising from LQR problems given in implicit form. In contrast to previously proposed schemes such as BDF or Rosenbrock methods, the splitting schemes exploit the fact that the nonlinear and affine parts of the problem, when considered in isolation, have closed-form solutions. We show that if the solution possesses low-rank structure, which is frequently the case, then this is preserved by the method. This feature is used to implement the methods efficiently for large-scale problems. The proposed methods are expected to be competitive, as they at most require the solution of a small number of linear equation systems per time step. Finally, we apply our low-rank implementations to the Riccati Equations arising from two LQR problems. The results show that the rank of the solutions stay low, and the expected orders of convergence are observed.
Patrick Kurschner - One of the best experts on this subject based on the ideXlab platform.
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a numerical comparison of different solvers for large scale continuous time algebraic Riccati Equations and lqr problems
SIAM Journal on Scientific Computing, 2020Co-Authors: Peter Benner, Patrick Kurschner, Zvonimir Bujanovic, Jens SaakAbstract:In this paper, we discuss numerical methods for solving large-scale continuous-time algebraic Riccati Equations. These methods have been the focus of intensive research in recent years, and signifi...
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a numerical comparison of solvers for large scale continuous time algebraic Riccati Equations and lqr problems
arXiv: Numerical Analysis, 2018Co-Authors: Peter Benner, Patrick Kurschner, Zvonimir Bujanovic, Jens SaakAbstract:In this paper, we discuss numerical methods for solving large-scale continuous-time algebraic Riccati Equations. These methods have been the focus of intensive research in recent years, and significant progress has been made in both the theoretical understanding and efficient implementation of various competing algorithms. There are several goals of this manuscript: first, to gather in one place an overview of different approaches for solving large-scale Riccati Equations, and to point to the recent advances in each of them. Second, to analyze and compare the main computational ingredients of these algorithms, to detect their strong points and their potential bottlenecks. And finally, to compare the effective implementations of all methods on a set of relevant benchmark examples, giving an indication of their relative performance.
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RADI: a low-rank ADI-type algorithm for large scale algebraic Riccati Equations
Numerische Mathematik, 2018Co-Authors: Peter Benner, Patrick Kurschner, Zvonimir Bujanović, Jens SaakAbstract:This paper introduces a new algorithm for solving large-scale continuous-time algebraic Riccati Equations (CARE). The advantage of the new algorithm is in its immediate and efficient low-rank formulation, which is a generalization of the Cholesky-factored variant of the Lyapunov ADI method. We discuss important implementation aspects of the algorithm, such as reducing the use of complex arithmetic and shift selection strategies. We show that there is a very tight relation between the new algorithm and three other algorithms for CARE previously known in the literature—all of these seemingly different methods in fact produce exactly the same iterates when used with the same parameters: they are algorithmically different descriptions of the same approximation sequence to the Riccati solution.
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low rank newton adi methods for large nonsymmetric algebraic Riccati Equations
Journal of The Franklin Institute-engineering and Applied Mathematics, 2016Co-Authors: Peter Benner, Patrick Kurschner, Jens SaakAbstract:Abstract The numerical treatment of large-scale, nonsymmetric algebraic Riccati Equations (NAREs) by a low-rank variant of Newton׳s method is considered. We discuss a method to compute approximations to the solution of the NARE in a factorized form of low rank. The occurring large-scale Sylvester Equations are dealt with using the factored alternating direction implicit iteration (fADI). Several performance enhancing strategies available for the factored ADI as well as the related Newton-ADI for symmetric algebraic Riccati Equations are generalized to this combination. This includes the efficient computation of the norm of the residual matrix, adapted shift parameter strategies for fADI, and an acceleration of Newton׳s scheme by means of a Galerkin projection. Numerical experiments illustrate the capabilities of the proposed method to solve high-dimensional NAREs.
Gerhard Jank - One of the best experts on this subject based on the ideXlab platform.
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matrix Riccati Equations in control and systems theory
2003Co-Authors: Hisham Aboukandil, Gerhard Freiling, Vlad Ionescu, Gerhard JankAbstract:1 Basic results for linear Equations.- 1.1 Linear differential Equations and linear algebraic Equations.- 1.2 Exponential dichotomy and L2evolutions.- 2 Hamiltonian Matrices and Algebraic Riccati Equations.- 2.1 Solutions of algebraic Riccati Equations and graph subspaces.- 2.2 Indefinite scalar products and a canonical form of Hamiltonian matrices.- 2.3 Hermitian algebraic Riccati Equations.- 2.4 Positive semi-definite solutions of standard algebraic Riccati Equations.- 2.5 Hermitian discrete-time algebraic Riccati Equations.- 3 Global aspects of Riccati differential and difference Equations.- 3.1 Riccati differential Equations and associated linear systems.- 3.1.1 Riccati differential Equations, Riccati-transformation and spectral factorization.- 3.1.2 Riccati differential Equations and linear boundary value problems.- 3.2 A representation formula.- 3.3 Flows on Grassmann manifolds: The extended Riccati differential equation.- 3.4 General representation formulae for solutions of RDE and PRDE, the time-continuous and periodic Riccati differential equation, and dichotomy.- 3.4.1 A general representation formula for solutions of RDE.- 3.4.2 A representation formula for solutions of the periodic Riccati differential equation PRDE.- 3.5 A representation formula for solutions of the discrete time Riccati equation.- 3.5.1 Properties of the solutions to DARE.- 3.5.2 Properties of the solutions to DRDE.- 3.6 Global existence results.- 4 Hermitian Riccati differential Equations.- 4.1 Comparison results for HRDE.- 4.1.1 Arbitrary coefficients.- 4.1.2 Periodic coefficients.- 4.1.3 Constant coefficients.- 4.1.4 Riccati inequalities.- 4.2 Monotonicity and convexity results: A Fre1 Basic results for linear Equations.- 1.1 Linear differential Equations and linear algebraic Equations.- 1.2 Exponential dichotomy and L2evolutions.- 2 Hamiltonian Matrices and Algebraic Riccati Equations.- 2.1 Solutions of algebraic Riccati Equations and graph subspaces.- 2.2 Indefinite scalar products and a canonical form of Hamiltonian matrices.- 2.3 Hermitian algebraic Riccati Equations.- 2.4 Positive semi-definite solutions of standard algebraic Riccati Equations.- 2.5 Hermitian discrete-time algebraic Riccati Equations.- 3 Global aspects of Riccati differential and difference Equations.- 3.1 Riccati differential Equations and associated linear systems.- 3.1.1 Riccati differential Equations, Riccati-transformation and spectral factorization.- 3.1.2 Riccati differential Equations and linear boundary value problems.- 3.2 A representation formula.- 3.3 Flows on Grassmann manifolds: The extended Riccati differential equation.- 3.4 General representation formulae for solutions of RDE and PRDE, the time-continuous and periodic Riccati differential equation, and dichotomy.- 3.4.1 A general representation formula for solutions of RDE.- 3.4.2 A representation formula for solutions of the periodic Riccati differential equation PRDE.- 3.5 A representation formula for solutions of the discrete time Riccati equation.- 3.5.1 Properties of the solutions to DARE.- 3.5.2 Properties of the solutions to DRDE.- 3.6 Global existence results.- 4 Hermitian Riccati differential Equations.- 4.1 Comparison results for HRDE.- 4.1.1 Arbitrary coefficients.- 4.1.2 Periodic coefficients.- 4.1.3 Constant coefficients.- 4.1.4 Riccati inequalities.- 4.2 Monotonicity and convexity results: A Frechet derivative based approach.- 4.2.1 Notation and preliminaries.- 4.2.2 Results for HARE.- 4.2.3 Results for HDARE.- 4.2.4 Results for HRDE.- 4.3 Convergence to the semi-stabilizing solution.- 4.4 Dependence of HRDE on a parameter.- 4.5 An existence theorem for general HRDE.- 4.6 A special property of HRDE.- 5 The periodic Riccati equation.- 5.1 Linear periodic differential Equations.- 5.2 Preliminary notation and results for linear periodic systems.- 5.3 Existence results for periodic Hermitian Riccati Equations.- 5.4 Positive semi-definite periodic equilibria of PRDE.- 6 Coupled and generalized Riccati Equations.- 6.1 Some basic concepts in dynamic games.- 6.2 Non-symmetric Riccati Equations in open loop Nash differential games.- 6.3 Discrete-time open loop Nash Riccati Equations.- 6.4 Non-symmetric Riccati Equations in open loop Stackelberg differential games.- 6.5 Discrete-time open loop Stackelberg Equations.- 6.6 Coupled Riccati Equations in closed loop Nash differential games.- 6.7 Rational matrix differential Equations arising in stochastic control.- 6.8 Rational matrix difference Equations arising in stochastic control.- 6.9 Coupled Riccati Equations in Markovian jump systems.- 7 Symmetric differential Riccati Equations: an operator based approach.- 7.1 Popov triplets: definition and equivalence.- 7.2 Associated objects.- 7.3 Associated operators.- 7.4 Existence of the stabilizing solution.- 7.5 Positivity theory and applications.- 7.6 Differential Riccati inequalities.- 7.7 The signature condition.- 7.8 Differential Riccati theory: A Hamiltonian descriptor operator approach.- 7.8.1 Descriptors and dichotomy.- 7.8.2 Hamiltonian descriptors.- 7.8.3 The stabilizing (anti-stabilizing) solution.- 8 Applications to Robust Control Systems.- 8.1 The Four Block Nehari Problem.- 8.1.1 Problem Statement.- 8.1.2 A characterization of all solutions.- 8.1.3 Main Result.- 8.2 Disturbance Attenuation.- 8.2.1 Problem statement.- 8.2.2 A necessary condition.- 8.2.3 The Disturbance Feedforward Problem.- 8.2.4 The least achievable tolerance of the DF problem.- 9 Non-symmetric Riccati theory and applications.- 9.1 Non-symmetric Riccati theory.- 9.1.1 Basic notions and preliminary results.- 9.1.2 Toeplitz operators and Riccati Equations.- 9.2 Application to open loop Nash games.- 9.2.1 Definitions and Hilbert space.- 9.2.2 Unique Nash equilibria.- 9.2.3 The general case.- 9.2.4 If any, then one or infinitely many.- 9.3 Application to open loop Stackelberg games.- 9.3.1 Characterization in Hilbert space.- 9.3.2 Unique Stackelberg equilibria.- 9.3.3 A value function type approach.- A Appendix.- A.1 Basic facts from control theory.- A.2 The implicit function theorem.- References.- List of Figures.
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coupled matrix Riccati Equations in minimal cost variance control problems
IEEE Transactions on Automatic Control, 1999Co-Authors: Gerhard Freiling, S R Lee, Gerhard JankAbstract:We present an algorithm for the solution of a nontrivial coupled system of algebraic Riccati Equations appearing in risk sensitive control problems. Moreover, we use comparison methods to derive non-blowup conditions for the solutions of a corresponding terminal value problem for coupled systems of Riccati differential Equations.
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solution and asymptotic behavior of coupled Riccati Equations in jump linear systems
IEEE Transactions on Automatic Control, 1994Co-Authors: Hisham Aboukandil, Gerhard Freiling, Gerhard JankAbstract:A new necessary and sufficient condition for the existence of a positive semidefinite solution of coupled Riccati Equations occurring in jump linear systems is derived. By verifying a Riccati inequality it is shown that such a solution exists; in addition two numerical algorithms are given to compute it. An example is given to illustrate the proposed method. >
