## Invited Session Tue.3.H 2035

#### Tuesday, 15:15 - 16:45 h, Room: H 2035

**Cluster 24: Variational analysis** [...]

### Control and optimization of impulsive systems II

**Chair: Dmitry Karamzin and Fernando Lobo Pereira**

**Tuesday, 15:15 - 15:40 h, Room: H 2035, Talk 1**

**Aram V. Arutyunov**

The R.V. Gamkrelidze's maximum principle for state constrained optimal control problem: Revisited

**Coauthors: Dmitry Karamzin, Fernando Lobo Pereira**

**Abstract:**

We study necessary conditions of optimality for optimal control problem with state constraints in the form of the Pontryagin's maximum principle (for short MP). For problems with state constraints these conditions were first obtained by R.V. Gamkrelidze in 1959 and subsequently published in the classic monograph by the four authors. This MP was obtained under a certain regularity assumption on the optimal trajectory. Somewhat later, in 1963, A.Ya. Dubovitskii and A.A. Milyutin proved another MP for problems with state constraints. In contrast with the MP of R.V. Gamkrelidze, this MP was obtained without a priori regularity assumptions, but it degenerates in many cases of interest what was dicovered and studied later. Here, we suggest a MP in the form proposed by R.V. Gamkrelidze without any a priori regularity assumptions on the optimal trajectory. However, without a priori regularity assumptions, this MP may degenerate. Therefore, we prove that, under certain additional conditions of controllability relatively to the state constraints at the end-points, or regularity of the control process, degeneracy will not occur, since a stronger non-triviality condition will be satisfied.

**Tuesday, 15:45 - 16:10 h, Room: H 2035, Talk 2**

**Elena Goncharova**

Impulsive systems with mixed constraints

**Coauthor: Maxim Staritsyn**

**Abstract:**

We consider an optimal control problem for an impulsive hybrid system. Such a dynamical system can be described by a nonlinear measure differential equation under mixed constraints on a state trajectory and a control measure. The

constraints are of the form

\begin{gather*}

Q_{-}\big(x(t-)\big) =0, Q_{+}\big(x(t)\big)=0, \

Ψ\big(x(t-)\big) ≤ 0, Ψ\big(x(t)\big) ≤ 0 ν -a.e. on *[0, T]. *

\end{gather*}

Here, *x(t-)*, *x(t)* are the left and right limits of a state trajectory *x* at time *t*, a non-negative scalar measure * ν * is the total variation of an "impulsive control'', and * ν ([0, T]) ≤ M* with *M>0*. Such conditions can be also regarded as state constraints of equality and inequality type qualified to hold only over the set where * ν * is localized. A time reparameterization technique is developed to establish a result on the problem transformation to a classical optimal control problem with absolutely continuous trajectories. Based on this result, a conceptual approach is proposed to design numerical methods for optimal impulsive control. We give some results on numerical simulation of a double pendulum with a blockable degree of freedom.

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**Tuesday, 16:15 - 16:40 h, Room: H 2035, Talk 3**

**Laurent Pfeiffer**

Sensitivity analysis for relaxed optimal control problems with final-state constraints

**Coauthors: Joseph Frédéric Bonnans, Oana Silvia Serea**

**Abstract:**

We consider a family of relaxed optimal control problems with final-state constraints, indexed by a perturbation variable *y*. Our goal is to compute a second-order expansion of the value *V(y)* of the problems, near a reference value of *y*. We use relaxed controls, i.e., the control variable is at each time a probability measure. Under some conditions, a constrained optimal control problem has the same value as its relaxed version.

The specificity of our study is to consider bounded strong solutions [2], i.e., local optimal solutions in a small neighborhood (for the *L*^{\}infty-distance) of the trajectory. To obtain a sharp second-order upper estimate of *V*, we derive two linearized problem from a wide class of perturbations of the control (e.g., small perturbations for the *L*^{1}-distance). Relaxation permits a very convenient linearization the problems. Using the decomposition principle [1], we prove that the upper estimate is an exact expansion.

J.F. Bonnans, N.P. Osmolovski\u\i. Second-order analysis of optimal control problems with control and final-state constraints. 2010.

- A.A. Milyutin, N.P. Osmolovski\u\i. Calculus of variations and optimal control. 1998.