Deterministic impulse control problems.

*(English)*Zbl 0571.49020The paper deals with the stationary optimal control problem of impulsive type. The available controls are continuous and impulsive and the controlled system is governed by ordinary differential equations in \(R^ N\). The main result of the paper states that the optimal cost function u associated to the control problem is the unique viscosity solution of the first-order Hamilton-Jacobi quasi-variational inequality (QVI) of the following form: \(\max (H(x,u,Du),u-Mu)=0\) in \(R^ N.\)

The principal analytical tools used are the dynamic programming principle, the theory of elliptic QVI and the concept of viscosity solution of Hamilton-Jacobi equations. In addition to the main result of existence and uniqueness, some properties of the optimal cost function u are proved (regularity, its behaviour at infinity and the fact that u is the maximum subsolution of the QVI in the distribution sense).

The principal analytical tools used are the dynamic programming principle, the theory of elliptic QVI and the concept of viscosity solution of Hamilton-Jacobi equations. In addition to the main result of existence and uniqueness, some properties of the optimal cost function u are proved (regularity, its behaviour at infinity and the fact that u is the maximum subsolution of the QVI in the distribution sense).

Reviewer: R.Gonzales

##### MSC:

49L20 | Dynamic programming in optimal control and differential games |

49J40 | Variational inequalities |

49L99 | Hamilton-Jacobi theories |

35D05 | Existence of generalized solutions of PDE (MSC2000) |

35D10 | Regularity of generalized solutions of PDE (MSC2000) |

35J85 | Unilateral problems; variational inequalities (elliptic type) (MSC2000) |