Short abstract: This paper concerns the relation between exact controllability and stabilizability with arbitrary decay rate in infinite dimensional spaces. It appears that in several cases the notions are equivalent, but there are a lot of situations when additional conditions are needed, for example in Banach spaces. This is a short and non exhaustive review of some research on control theory for infinite dimensional spaces initiated by V.I. Korobov during the 70th of the past century in Kharkiv State University.
Extended abstract: In this paper we consider linear systems with control described by the equation $\dot x = A x +B u$ where functions $u$ and $x$ take values in $U$ and $X$ respectively. For such object, a short review of results concerning relations between exact controllability and complete stabilizability (stabilizability with arbitrary decay rate) is given. The analysis is done in various situations: bounded or unbounded state and control operators $A$ and $B$, Banach or Hilbert spaces $U$ and $X$. The well known equivalence between complete controllability and pole assignment in the situation of finite dimensional spaces is no longer true in general in infinite dimensional spaces. Exact controllability is not sufficient for complete stabilizability if $U$ and $X$ are Banach spaces. In Hilbert space setting this implication holds true. The converse also is not so simple: in some situations, complete stabilizability implies exact controllability (Banach space setting with bounded operators), in other situation, it is not true. The corresponding results are given with some ideas for the proofs. Complete technical development are indicated in the cited literature. Several examples are given. Special attention is paid to the case of infinite dimensional systems generated by delay systems of neutral type in some general form (distributed delays). The question of the relation between exact null controllability and complete stabilizability is more precisely investigated. In general there is no equivalence between the two notions. However for some classes of neutral type equations there is an equivalence. The question how the equivalence occurs for more general systems is still open. This is a short and non exhaustive review of some research on control theory for infinite dimensional spaces. Our works in this area were initiated by V.I. Korobov during the 70th of the past century in Kharkiv State University.
Keywords: Exact controllability; complete stabilizability; infinite dimensional systems; neutral type.
2010 Mathematics Subject Classification: 93B05; 93D15; 93C23; 93C43. [ Full-text available (PDF) ] Top of the page.
Short abstract: The controllability function method, introduced by V.I. Korobov in late 1970s, is known to be an efficient tool for control systems design. This paper bridges the method with the homogeneity theory popular today. In particular, it is shown that the so-called homogeneous norm is a controllability function of the system in some cases. Moreover, the closed-loop control system is homogeneous in a generalized sense. This immediately yields many useful properties of the system such as robustness (Input-to-State Stability) with respect to a rather large class of perturbations.
Extended abstract: The controllability function method, introduced by V.I. Korobov in late 1970s, is known to be an efficient tool for control systems design. It is developed for both linear/nonlinear and finite/infinite dimensional systems. This paper bridges the method with the homogeneity theory popular today. The standard homogeneity known since 18th century is a symmetry of function with respect to uniform scaling of its argument. Some generalizations of the standard homogeneity were introduced in 20th century. This paper shows that the so-called homogeneous norm is a controllability function of the linear autonomous control system and the corresponding closed-loop system is homogeneous in the generalized sense. This immediately yields many useful properties known for homogeneous systems such as robustness (Input-to-State Stability) with respect to a rather large class of perturbations, in particular, with respect to bounded additive measurement noises and bounded additive exogenous disturbances. The main theorem presented in this paper slightly refines the design of the controllability function for a multiply-input linear autonomous control systems. The design procedure consists in solving subsequently a linear algebraic equation and a system of linear matrix inequalities. The homogeneity itself and the use of the canonical homogeneous norm essentially simplify the design of a controllability function and the analysis of the closed-loop system. Theoretical results are supported with examples. The further study of homogeneity-based design of controllability functions seems to be a promising direction for future research.
Keywords: controllability function; generalized homogeneity; robustness.
2010 Mathematics Subject Classification: 34H05; 34H15; 93C10; 93C15. [ Full-text available (PDF) ] Top of the page.
Short abstract: We prove a small-gain sufficient condition for (global) finite-time input-to-state stability (FTISS) of infinite networks. The network under consideration is composed of a countable set of finite-dimensional subsystems of ordinary differential equations, each of which is interconnected with a finite number of its ``neighbors'' only and is affected by some external disturbances. We assume that each node (subsystem) of our network is finite-time input-to-state stable (FTISS) with respect to its finite-dimensional inputs produced by this finite set of the neighbors and with respect to the corresponding external disturbance. As an application we obtain a new theorem on decentralized {\em finite-time} input-to-state stabilization with respect to external disturbances for infinite networks composed of a countable set of strict-feedback form systems of ordinary differential equations. For this we combine our small-gain theorem proposed in the current work with the controllers design developed by S.Pavlichkov and C.K.Pang (NOLCOS-2016) for the gain assignment of the strict-feedback form systems in the case of finite networks. The current results address the FTISS and decentralized FTISS stabilization and redesign the technique proposed in recent work S. Dashkovskiy and S. Pavlichkov, Stability conditions for infinite networks of nonlinear systems and their application for stabilization, Automatica. - 2020. -112. -108643, in which the case of $\ell_{\infty}$-ISS of infinite networks was investigated.
Extended abstract: We prove a small-gain sufficient condition for (global) finite-time input-to-state stability (FTISS) of infinite networks. The network under consideration is composed of a countable set of finite-dimensional subsystems of ordinary differential equations, each of which is interconnected with a finite number of its ``neighbors'' only and is affected by some external disturbances. We assume that each node (subsystem) of our network is finite-time input-to-state stable (FTISS) with respect to its finite-dimensional inputs produced by this finite set of the neighbors and with respect to the corresponding external disturbance. As an application we obtain a new theorem on decentralized {\em finite-time} input-to-state stabilization with respect to external disturbances for infinite networks composed of a countable set of strict-feedback form systems of ordinary differential equations. For this we combine our small-gain theorem proposed in the current work with the controllers design developed by S. Pavlichkov and C.K. Pang (NOLCOS-2016) for the gain assignment of the strict-feedback form systems in the case of finite networks. The current results address the finite-time input-to-state stability and decentralized finite-time input-to-state stabilization and redesign the technique proposed in recent work S. Dashkovskiy and S. Pavlichkov, Stability conditions for infinite networks of nonlinear systems and their application for stabilization, Automatica. -2020. -112. -108643, in which the case of $\ell_{\infty}$-ISS of infinite networks was investigated. The current paper extends and generalizes its conference predecessor to the case of finite-time {\em ISS} stability and decentralized stabilization in presence of external disturbance inputs and with respect to these disturbance inputs. In the special case when all these external disturbances are zeroes (i.e. are abscent), we just obtain finite-time stability and finite-time decentralized stabilization of infinite networks accordingly.
Keywords: nonlinear systems; input-to-state stability; small gain conditions.
2010 Mathematics Subject Classification: 93C10; 93A15; 93D25; 93B70; 93D40; 93A14. [ Full-text available (PDF) ] Top of the page.
Short abstract: In this article we found the conditions of the existence and constructive scheme for finding the solutions of the linear Noetherian differential-algebraic boundary value problem for a differential-algebraic equation with singular impulse action.
Extended abstract: The study of differential-algebraic boundary value problems was initiated in the works of K. Weierstrass, N. N. Luzin and F. R. Gantmacher. Systematic study of differential-algebraic boundary value problems is devoted to the work of S. Campbell, Yu. E. Boyarintsev, V. F. Chistyakov, A. M. Samoilenko, M. O. Perestyuk, V. P. Yakovets, O. A. Boichuk, A. Ilchmann and T. Reis. The study of the differential-algebraic boundary value problems is associated with numerous applications of such problems in the theory of nonlinear oscillations, in mechanics, biology, radio engineering, theory of control, theory of motion stability. At the same time, the study of differential algebraic boundary value problems is closely related to the study of pulse boundary value problems for differential equations, initiated M. O. Bogolybov, A. D. Myshkis, A. M. Samoilenko, M. O. Perestyk and O. A. Boichuk. Consequently, the actual problem is the transfer of the results obtained in the articles by S. Campbell, A. M. Samoilenko, M. O. Perestyuk and O. A. Boichuk on a pulse linear boundary value problems for differential-algebraic equations, in particular finding the necessary and sufficient conditions for the existence of the desired solutions, and also the construction of the Green's operator of the Cauchy problem and the generalized Green operator of a pulse linear boundary value problem for a differentialalgebraic equation. In this article we found the conditions of the existence and constructive scheme for finding the solutions of the linear Noetherian differential-algebraic boundary value problem for a differential-algebraic equation with singular impulse action. The proposed scheme of the research of the linear differential-algebraic boundary value problem for a differential-algebraic equation with impulse action in the critical case in this article can be transferred to the linear differential-algebraic boundary value problem for a differentialalgebraic equation with singular impulse action. The above scheme of the analysis of the seminonlinear differential-algebraic boundary value problems with impulse action neralizes the results of S. Campbell, A. M. Samoilenko, M. O. Perestyuk and O. A. Boichuk and can be used for proving the solvability and constructing solutions of weakly nonlinear boundary value problems with singular impulse action in the critical and noncritical cases.
Keywords: differential-algebraic equations; boundary value problems; pulse influence.
2010 Mathematics Subject Classification: 15A24; 34B15; 34C25. [ Full-text available (PDF) ] Top of the page.
Short abstract: The article is devoted to the problem of wave-packet propagation in a three - layer hydrodynamic system "layer with a hard bottom - layer - layer with a cover", stratified by density. Using the method of multiscale developments, the first three approximations of the studied problem are obtained, of which the first two are given in the article. The solutions of the first approximation and the variance relation are presented. The evolution equations of the circumferential wave packets on the contact surfaces are derived in the form of the nonlinear Schrodinger equation. A partial solution of the nonlinear Schrodinger equation is obtained. The formulas of deviations of contact surfaces and the conditions under which the shape of wave-packets on the upper and lower contact surfaces changes are derived. The regions of familiarity of the coefficients for the second harmonics on the upper and lower contact surfaces for both frequency pairs are given and analyzed. Various cases in which there is an asymmetry in the shape of wave-packets also graphically illustrated and analyzed.
Extended abstract: The article is devoted to the problem of wave-packet propagation in a three - layer hydrodynamic system "layer with a hard bottom - layer - layer with a cover", stratified by density. The current research on selected topics is reviewed. The mathematical formulation of the problem is given in dimensionless form and contains the equations of fluid motion, kinematic and dynamic conditions on the contact surfaces, as well as the boundary conditions on the lid and on the bottom. Using the method of multiscale developments, the first three approximations of the studied problem are obtained, of which the first two are given in the article, because the third approximation has a cumbersome analytical form. The solutions of the first approximation and the variance relation are presented. The evolution equations of the circumferential wave-packets on the contact surfaces are derived in the form of the nonlinear Schrodinger equation on the basis of the variance relation and the conditions for the solvability of the second and third approximations. A partial solution of the nonlinear Schrodinger equation is obtained after the transition to a system moving with group velocity. For the first and second approximations, the formulas for the deviations of the contact surfaces are derived, taking into account the solution of the nonlinear Schrodinger equation. The conditions under which the shape of wave-packets on the upper and lower contact surfaces changes are derived. The regions of familiarity of the coefficients for the second harmonics on the upper and lower contact surfaces for both frequency pairs, which are the roots of the variance relation, are presented and analyzed. Also, for both frequency pairs, different cases of superimposition of maxima and minima of the first and second harmonics, in which there is an asymmetry in the shape of wave packets, are graphically illustrated and analyzed. All results are illustrated graphically. Analytical transformations, calculations and graphical representation of results were performed using a package of symbolic calculations and computer algebra Maple.
Keywords: three-layer hydrodynamic system; wave-packets; shape of wave-packets.
2010 Mathematics Subject Classification: 76A02; 76B15; 76M35. [ Full-text available (PDF) ] Top of the page.
Short abstract: September 27, 2021 turned 80 years since the birth of Professor Valerii Ivanovich Korobov. The editorial board welcomes.
2010 Mathematics Subject Classification: 01A70. [ Full-text available (PDF) ] Top of the page.