Chandrasekhar equations for infinite dimensional systems. Part 2: Unbounded input and output case

A set of equations known as Chandrasekhar equations arising in the linear quadratic optimal control problem is considered. In this paper, we consider the linear time-invariant system defined in Hilbert spaces involving unbounded input and output operators. For a general class of such systems, the Chandrasekhar equations are derived and the existence, uniqueness, and regularity of the results of their solutions established

Chandrasekhar equations for infinite dimensional systems. II. Unbounded input and output case

AbstractA set of equations known as Chandrasekhar equations arising in the linear quadratic optimal control problem is considered. In this paper, we consider the linear time-invariant systems defined in Hilbert spaces involving unbounded input and output operators. For a general class of such systems, we derive the Chandrasekhar equations and establish the existence, uniqueness, and regularity results of their solutions

Chandrasekhar equations for infinite dimensional systems

Chandrasekhar equations are derived for linear time invariant systems defined on Hilbert spaces using a functional analytic technique. An important consequence of this is that the solution to the evolutional Riccati equation is strongly differentiable in time and one can define a strong solution of the Riccati differential equation. A detailed discussion on the linear quadratic optimal control problem for hereditary differential systems is also included

Frequency theorem for the regulator problem with unbounded cost functional and its applications to nonlinear delay equations

We study the quadratic regulator problem with unbounded cost functional in a Hilbert space. A motivation comes from delay equations, which has the feedback part with discrete delays (or, in other words, delta-like measurements, which are unbounded in $L_{2}$). We treat the problem in an abstract context of a Hilbert space, which is rigged by a Banach space. We obtain a version of the Frequency Theorem, which guarantees the existence of a unique optimal process (starting in the Banach space) and shows that the optimal cost is given by a quadratic Lyapunov-like functional. In our adjacent works it is shown that these functionals can be used to construct inertial manifolds for delay equations and allow to treat and extend many papers in the field of applied dynamics (especially, developments of convergence theorems and the Poincar\'{e}-Bendixson theory done by R. A. Smith, construction of inertial manifolds for delay equations with small delays done by C. Chicone, Yu. A. Ryabov and R. D. Driver) in a unified manner. We also present more concrete applications concerned with frequency-domain stability criteria, which in particular cases coincide with the well-known Circle Criterion

Geometric theory of inertial manifolds for compact cocycles in Banach spaces

We present a geometric theory of inertial manifolds for compact cocycles (non-autonomous dynamical systems), which satisfy a certain squeezing property with respect to a family of quadratic Lyapunov functionals in a Banach space. Under general assumptions we show that these manifolds posses classical properties such as exponential tracking, differentiability and normal hyperbolicity. Our theory includes and largely extends classical studies for semilinear parabolic equations by C. Foias, G. R. Sell and R. Temam (based on the Spectral Gap Condition) and by G. R. Sell and J. Mallet-Paret (based on the Spatial Averaging) and their further developments. Besides semilinear parabolic equations our theory can be applied also to ODEs, ODEs with delay (extending the inertial manifold theories of Yu. A. Ryabov, R. D. Driver and C. Chicone for delay equations with small delays), parabolic equations with delay and parabolic equations with boundary controls (nonlinear boundary conditions). In applications, the squeezing property can be verified with the aid of various versions of the Frequency Theorem, which provides optimal (in some sense) and flexible conditions. This flexibility gives a possibility in applications to obtain conditions for low-dimensional dynamics, including, in particular, developments of the Poincar\'{e}-Bendixson theory and convergence theorems from a series of papers by R. A. Smith

Strict Lyapunov functions for semilinear parabolic partial differential equations

International audienceFor families of partial differential equations (PDEs) with particular boundary conditions, strict Lyapunov functions are constructed. The PDEs under consideration are parabolic and, in addition to the diffusion term, may contain a nonlinear source term plus a convection term. The boundary conditions may be either the classical Dirichlet conditions, or the Neumann boundary conditions or a periodic one. The constructions rely on the knowledge of weak Lyapunov functions for the nonlinear source term. The strict Lyapunov functions are used to prove asymptotic stability in the framework of an appropriate topology. Moreover, when an uncertainty is considered, our construction of a strict Lyapunov function makes it possible to establish some robustness properties of Input-to-State Stability (ISS) type

Scalar evolution equations for shear waves in incompressible solids: A simple derivation of the Z, ZK, KZK, and KP equations

We study the propagation of two-dimensional finite-amplitude shear waves in a nonlinear pre-strained incompressible solid, and derive several asymptotic amplitude equations in a simple, consistent, and rigorous manner. The scalar Zabolotskaya (Z) equation is shown to be the asymptotic limit of the equations of motion for all elastic generalized neo-Hookean solids (with strain energy depending only on the first principal invariant of Cauchy-Green strain). However, we show that the Z equation cannot be a scalar equation for the propagation of two-dimensional shear waves in general elastic materials (with strain energy depending on the first and second principal invariants of strain). Then we introduce dispersive and dissipative terms to deduce the scalar Kadomtsev-Petviashvili (KP), Zabolotskaya-Khokhlov (ZK) and Khokhlov-Zabolotskaya-Kuznetsov (KZK) equations of incompressible solid mechanics