TY - JOUR

T1 - Frequency theorem and inertial manifolds for neutral delay equations

AU - Аникушин, Михаил Михайлович

PY - 2023/12/1

Y1 - 2023/12/1

N2 - We study the infinite-horizon quadratic regulator problem for linear control systems in Hilbert spaces, where the cost functional is in some sense unbounded. Our motivation comes from delay equations with the feedback part containing discrete delays or, in other words, measurements given by δ -functionals, which are unbounded in L2 . Working in an abstract context in which such (and many others, including parabolic boundary control problems) equations can be treated, we obtain a version of the Frequency Theorem. It guarantees the existence of a unique optimal process and shows that the optimal cost is given by a quadratic Lyapunov-like functional. In our adjacent works it is shown that such functionals can be used to construct inertial manifolds and allow to treat and extend many works in the field in a unified manner. Here we concentrate on applications to delay equations and especially mention the works of R.A. Smith on developments of convergence theorems and the Poincaré-Bendixson theory; and also the works of Yu.A. Ryabov, R.D. Driver and C. Chicone on inertial manifolds for equations with small delays and their recent generalization for equations of neutral type given by S. Chen and J. Shen.

AB - We study the infinite-horizon quadratic regulator problem for linear control systems in Hilbert spaces, where the cost functional is in some sense unbounded. Our motivation comes from delay equations with the feedback part containing discrete delays or, in other words, measurements given by δ -functionals, which are unbounded in L2 . Working in an abstract context in which such (and many others, including parabolic boundary control problems) equations can be treated, we obtain a version of the Frequency Theorem. It guarantees the existence of a unique optimal process and shows that the optimal cost is given by a quadratic Lyapunov-like functional. In our adjacent works it is shown that such functionals can be used to construct inertial manifolds and allow to treat and extend many works in the field in a unified manner. Here we concentrate on applications to delay equations and especially mention the works of R.A. Smith on developments of convergence theorems and the Poincaré-Bendixson theory; and also the works of Yu.A. Ryabov, R.D. Driver and C. Chicone on inertial manifolds for equations with small delays and their recent generalization for equations of neutral type given by S. Chen and J. Shen.

KW - Frequency theorem

KW - Delay equations

KW - Inertial manifolds

KW - Lyapunov functionals

KW - Delay equations

KW - Frequency theorem

KW - Inertial manifolds

KW - Lyapunov functionals

UR - https://www.mendeley.com/catalogue/406dac69-dedc-31cb-899b-6516f3229b10/

U2 - 10.1007/s00028-023-00915-w

DO - 10.1007/s00028-023-00915-w

M3 - Article

VL - 23

JO - Journal of Evolution Equations

JF - Journal of Evolution Equations

SN - 1424-3199

IS - 4

M1 - 66

ER -