##
**Dynamic impulse systems: theory and applications.**
*(English)*
Zbl 0880.46031

Mathematics and its Applications (Dordrecht). 394. Dordrecht: Kluwer. xi, 256 p. (1997).

After a general presentation of Schwartz’s distributions in the first chapter, in Chapter 2 one expounds some aspects concerning the investigation of equations in distributions based on an integral of the distributions and on the multiplication of distributions by ordinary (in particular discontinuous) functions.

Using the notion of distribution depending on a parameter and a Lebesgue construction of the integral with respect to the parameter one obtains an integral form of a theorem on the structure of distributions with finite order of singularity which makes possible to derive a formula of Cauchy-Duhamel type for differential equations in distributions, a new representation of an impulse transfer function and a formula of Cauchy-Green type for a non-stationary convolution equation.

Chapter 3 deals with applications to problems of dynamics and control; particularly one gives a topological characterization of the concept of stability, presents schemes for solving singular problems of quadratic optimization of linear systems respectively for finding discontinuous periodic motions of systems with automatic regulation and formulates necessary conditions for optimality in the Lagrange problem in L. S. Pontryagin’s form with differential constraints containing product of discontinuous functions and impulse function.

Using these results, a lot of optimization problems are considered in Chapter 4. Among them the construction of programs for the displacement of manipulators and cylinders that minimize the energy costs of overcoming the force of resistance of a viscous medium, extremals in a problem on optimal interorbital flights, the construction of potential fields that correspond to the minimal value of the kinetic energy of a micro-object whose wave function satisfies the stationary Schrödinger equation but also mathematical models for discontinuous present price, the stability of equilibrium in general case and singular market optimization problems.

In Chapter 5 one considers differential equations containing products of discontinuous functions and distributional derivatives of functions of bounded variation whose solutions are taken as limits of sequences of ordinary solutions when the functions of bounded variation are replaced by approximating sequences of absolutely continuous functions. Particularly, a generalization of Gronwall-Bellman’s lemma for the space of functions of bounded variation is given.

In the last chapter, one studies the attainability set for a dynamic system with impulsive integrally bounded control and possible ways to construct the attainability set. One proves that this set is compact, connected and is continuously dependent on parameters and a control resource.

Let me underline the fact that the greatest part of the results presented in this book belong to the two authors and have been published in almost 40 papers in different mathematical journals.

For a review of the Russian original (Moscow Nauka, 1991), see Zbl 0745.47042.

Using the notion of distribution depending on a parameter and a Lebesgue construction of the integral with respect to the parameter one obtains an integral form of a theorem on the structure of distributions with finite order of singularity which makes possible to derive a formula of Cauchy-Duhamel type for differential equations in distributions, a new representation of an impulse transfer function and a formula of Cauchy-Green type for a non-stationary convolution equation.

Chapter 3 deals with applications to problems of dynamics and control; particularly one gives a topological characterization of the concept of stability, presents schemes for solving singular problems of quadratic optimization of linear systems respectively for finding discontinuous periodic motions of systems with automatic regulation and formulates necessary conditions for optimality in the Lagrange problem in L. S. Pontryagin’s form with differential constraints containing product of discontinuous functions and impulse function.

Using these results, a lot of optimization problems are considered in Chapter 4. Among them the construction of programs for the displacement of manipulators and cylinders that minimize the energy costs of overcoming the force of resistance of a viscous medium, extremals in a problem on optimal interorbital flights, the construction of potential fields that correspond to the minimal value of the kinetic energy of a micro-object whose wave function satisfies the stationary Schrödinger equation but also mathematical models for discontinuous present price, the stability of equilibrium in general case and singular market optimization problems.

In Chapter 5 one considers differential equations containing products of discontinuous functions and distributional derivatives of functions of bounded variation whose solutions are taken as limits of sequences of ordinary solutions when the functions of bounded variation are replaced by approximating sequences of absolutely continuous functions. Particularly, a generalization of Gronwall-Bellman’s lemma for the space of functions of bounded variation is given.

In the last chapter, one studies the attainability set for a dynamic system with impulsive integrally bounded control and possible ways to construct the attainability set. One proves that this set is compact, connected and is continuously dependent on parameters and a control resource.

Let me underline the fact that the greatest part of the results presented in this book belong to the two authors and have been published in almost 40 papers in different mathematical journals.

For a review of the Russian original (Moscow Nauka, 1991), see Zbl 0745.47042.

Reviewer: P.Craciunas (Iaşi)

### MSC:

46F10 | Operations with distributions and generalized functions |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

47E05 | General theory of ordinary differential operators |

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

49J27 | Existence theories for problems in abstract spaces |

93C25 | Control/observation systems in abstract spaces |