# Optimal Impulse Control of a Simple Reparable System in a Nonreflexive Banach Space

## TL;DR

This paper develops a mathematical framework for optimal impulse control in a reparable system modeled by complex equations in a nonreflexive Banach space, focusing on minimizing failure probability through state-dependent controls.

## Contribution

It introduces a novel approach to impulse control in a nonreflexive Banach space setting, proving existence and deriving optimality conditions for the control problem.

## Key findings

- Existence of an optimal impulse controller is rigorously proven.
- First-order necessary conditions for optimality are derived using variational inequalities.
- Control inputs are designed to depend on system state to ensure nonnegativity.

## Abstract

We discuss the problem of optimal impulse control representing the preventive maintenance of a simple reparable system. The system model is governed by coupled transport and integro-differential equations in a nonreflexive Banach space. The objective of this paper is to construct nonnegative impulse control inputs at given system running times that minimize the probability of the system in failure mode. To guarantee the nonnegativity of the controlled system, we consider the control inputs to depend on the system state. This essentially leads to a bilinear control problem. We first present a rigorous proof of existence of an optimal controller and then apply the variational inequality to derive the first-order necessary conditions of optimality.