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Technische Universit at Chemnitz Technische Universitat Chemnitz Sonderforschungsbereich Numerische Simulation auf massiv parallelen Rechnern Peter Benner Ralph Byers Volker Mehrmann Hongguo Xu Numerical Computation of Deating Subspaces of Embedded Hamiltonian Pencils Preprint SFB PreprintReihe des Chemnitzer SFB SFB June Contents Introduction and Preliminaries Embedding of Extended Matrix Pencils Hamiltonian Triangular Forms SkewHamiltonianHamiltonian Matrix Pencils Algorithms Error and Perturbation Analysis Conclusion A Algorithmic Details A Schur Forms for SkewHamiltonianHamiltonian Matrix Pencils and SkewHa miltonianskewHamiltonian Matrix Pencils A Subroutines Required by Algorithm Authors Peter Benner Ralph Byers Zentrum f ur Technomathematik Department of Mathematics Fachbereich Mathematik und Informatik University of Kansas Universitat Bremen Lawrence KS D Bremen FRG USA bennermathunibremende byersmathukansedu Volker Mehrmann Hongguo Xu Deparment of Mathematics Fakultat f ur Mathematik Case Western University Technische Universitat Chemnitz Cleveland OH D Chemnitz FRG USA mehrmannmathematiktuchemnitzde hxxpocwruedu Numerical Computation of Deating Subspaces of Embedded Hamiltonian Pencils 2 3 Peter Benner Ralph Byers Volker Mehrmann Hongguo Xu Abstract We discuss the numerical solution of structured generalized eigenvalue problems that arise from linearquadratic optimal control problems H optimization multibo dy sys 1 tems and many other areas of applied mathematics physics and chemistry The classical approach for these problems requires computing invariant and deating subspaces of ma trices and matrix p encils with Hamiltonian andor skewHamiltonian structure We ex tend the recently developed metho ds for Hamiltonian matrices and matrix p encils to the general case of embedded matrix p encils The rounding error and p erturbation analysis of the resulting algorithms is favorable Keywords eigenvalue problem deating subspace algebraic Riccati equation Hamiltonian matrix skewHamiltonian matrix skewHamiltonianHamiltonian matrix p encil embedded matrix p encils AMS sub ject classication N F B B C Introduction and Preliminaries In this pap er we study eigenvalue and invariant subspace computations involving matrices and matrix p encils with the following algebraic structures i h I 0 n where I is the n n identity matrix Denition Let J n I 0 n 2n2n H a A matrix H C is Hamiltonian if HJ HJ The Lie Algebra of Hamiltonian 2 n2n matrices in C is denoted by H 2n 2n2n H b A matrix H C is skewHamiltonian if HJ H J The Jordan algebra of 2 n2n skewHamiltonian matrices in C is denoted by SH 2n 2 n2n c A matrix pencil S H C is skewHamiltonianHamiltonian if S SH and 2n H H 2n 2 n2n H d A matrix S C is symplectic if SJ S J The Lie group of symplectic matrices 2 n2n in C is denoted by S 2n H 2n2n H I The J and U U e A matrix U C is unitary symplectic if U J U 2n d d d d d 2 n2n compact Lie group of unitary symplectic matrices in C is denoted by US 2n 2n H f A subspace L of C is called Lagrangian if it has dimension n and x J y for al l x y L Working title was Numerical Computation of Deating Subspaces of Embedded Hamiltonian and Sym plectic Pencils All authors were partially supp orted by Deutsche Forschungsgemeinschaft Research Grant Me and Sonderforschungsbereich Numerische Simulation auf massiv parallelen Rechnern 2 This author was partially supp orted by National Science Foundation awards CCR MRI and by the NSF EPSCoRKSTAR program through the Center for Advanced Scientic Computing 3 This work was completed while this author was with the TU Chemnitz There is little dierence b etween the structure of complex skewHamiltonian matrices and complex Hamiltonian matrices If S SH and H H then iS H and iH 2n 2 n 2n SH Similarly there is little dierence b etween the structure of complex skewHamiltoni 2n anHamiltonian matrix p encils complex skewHamiltonianskewHamiltonian and complex HamiltonianHamiltonian matrix p encils For S SH and H H the matrix p encil 2n 2 n S H is skewHamiltonianHamiltonian the matrix p encil S iH is skewHamil tonianskewHamiltonian and the matrix p encil iS H is HamiltonianHamiltonian Here i and i However real skewHamiltonian matrices are not real scalar multiples of Hamiltonian matrices There is a greater dierence in the structure of real skewHamiltonian and real Hamiltonian matrices than in the complex case The structures in Denition arise in linearquadratic optimal control H optimization and several other areas of applied mathematics computational physics and chemistry eg gyroscopic systems numerical simulation of elastic deformation and linear resp onse theory Here we fo cus on applications in linearquadratic optimal control and H optimization First we consider the continuous time innite horizon linearquadratic optimal control problem Choose a control function ut to minimize the cost functional Z H xt Q S xt S dt c H ut S R ut t 0 subject to the linear dierentialalgebraic system descriptor control system 0 E x Ax B u xt x 0 m n nn nm H nn H mm Here ut C xt C A E C B C Q Q C R R C and nm S C If the m n m n weighting matrix Q S R H S R is Hermitian and p ositive semidenite then the problem is wellposed Note that R Q andor R may b e singular Typically in addition to minimizing the control ut must make xt asymptotically stable Of course if R is p ositive denite then asymptotic stability of xt is automatically achieved by any control for which the cost functional is nite Application of the maximum principle yields as a necessary optimality condition that the control u satises the twopoint b oundary value problem of EulerLagrange equations x x 0 H E A xt x lim E t c c 0 t u u with the matrix p encil A B E H H Q A S E E A c c H H S B R The matrix p encil do es not have matrices with one of the structures of Denition Nevertheless many of the prop erties of Hamiltonian matrices carry over As reviewed b e low if E andor R are nonsingular then and reduce to an ordinary dierential equation with Hamiltonian matrix co ecients or a dierentialalgebraic equation with a skewHamilto nianHamiltonian matrix p encil co ecients In Section we circumvent the nonsingularity assumptions by embedding the matrix p encil into a skewHamiltonianHamiltonian ma trix p encil of larger dimension H If b oth E and R are nonsingular then with E reduces to the twopoint b oundary value problem x x 0 H xt x lim t 0 t with the Hamiltonian matrix 1 1 H 1 1 H H F G E A BR S E BR B E H 1 H 1 1 H H H Q SR S E A BR S H F This is the classical formulation found in many textb o oks on linearquadratic optimal control like The solution of this b oundary value problem can b e obtained in many dierent ways For example let Y b e a symmetric solution if it exists of the asso ciated continuoustime algebraic Riccati equation H H Y F F Y Y GY h i h i h i h i I 0 x I 0 x Multiplying from the left by the matrix and changing variables to Y I Y I one obtains the decoupled system x F GY G x xt x lim t 0 0 H t F Y G In the desired solution is identically zero In that case x is the solution to x F GY x 0 H with initial condition x x Y x E and the control that minimizes 1 H H H is u R S B E Y x If E is singular and R is nonsingular then do es not simplify quite so much b ecause it is a dierentialalgebraic equation with nontrivial linear constraints Substituting ut 1 H H R S xt B t system do es simplify to x x 0 H S H xt x lim E t 0 t with the reduced skewHamiltonianHamiltonian matrix p encil 1 H 1 H E A BR S BR B S H H 1 H 1 H E SR S Q A BR S In this case the corresp onding generalized algebraic Riccati equation is 1 H H 1 Q SR S E Y A BR S 1 H H 1 H A BR S Y E E Y BR B Y E but in general the relationship with solutions of the optimal control problem is lost or hidden See for details But even for E nonsingular the formulation in is preferable from a numerical p oint of view if E is illconditioned with resp ect to inversion If R is singular and E is nonsingular then the situation b ecomes still more complicated Although the b oundary value problem remains well dened the Riccati equation do es not This case has b een recently studied At this writing the case in which b oth E and R are singular has not b een analyzed in full generality The EulerLagrange and Riccati equations their solvability and their numerical solution have b een the sub ject of numerous publications in recent years see eg and the references therein In most numerical metho ds the Riccati solutions are obtained through the computation of deating or invariant subspaces of asso ciated matrix p encils see A key observation in this context is that the Riccati solutions need not b e formed explicitly It suces to work with bases of the deating subspaces of the matrix p encil For example supp ose E A in has an ndimensional deating subspace asso ciated c c with eigenvalues in the left half plane This can only b e the case if E is invertible Let this subspace b e spanned by the columns of a matrix U partitioned conformally with as U 1 U U 2 U 3 1 If U is invertible then the optimal control is a linear feedback of the form ut U U xt 1 3 1 1 1 and the solution of the asso ciated Riccati equation is Y U U E see Observe 2 1 that the optimal control may
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