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Discrete Time Riccati Equations and Invariant Subspaces of Linear Operators

机译：离散时间Riccati方程和线性算子的不变子空间

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Let U, H and Y be separable Hilbert spaces. Let A be a member of L(H), B be a211u001emember of L(U;H), C be a member of L(H;Y), and D be a member of L(U;Y) such that 211u001ethe open loop transfer function D(z):=D+zC(I-zA)(sup-1)B is a member of H(sup 211u001einfinity) (L(U;Y)). Let J>=0 be a self-adjoint cost operator. The authors study a 211u001esubset of self-adjoint solutions P of the discrete time algebraic Riccati 211u001eequation (DARE) A(sup asterisk)PA-P+C(sup astrisk)JC=K(sup asterisk)(sub 211u001eP)Lambda(sup P), Lambda(sub P)=D(sup asterisk)JD+B(sup astersk)PB, Lambda(sub 211u001eP)K(sub P)=-D(sup aserisk)JC-B(sup asterisk)PA, where Lambda(Sub P), Lambda(sup-211u001e1)(sub P) is a member of L(U)and K(sub P) is amember of L(H;U). The authors 211u001efurther assume that a critical solution P(sup crit) of DARE exists, such that 211u001eX(z):=1=zK(sub P(sub crit))(I-z(A+BK(sub P(sub crit))))(sup-1)B is a member of 211u001eH(sup infinity)(L(U;Y)) become an outer factor of D(z). Under technical 211u001eassumptions, the authors study connections of the nonnegative solution of DARE to 211u001ethe invariant subspace structure of (A(sup crit))(sup asterisk).

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