Subject Ph. D.

** Abstract: Future generations of safety-critical control systems will have to operate under multiple constraints. These constraints concern both the internal states, which must remain within admissible ranges (for example, biological concentrations, fuel levels, or battery charge must remain positive), and the actuators, which are always subject to saturation limits. The overall control resources (energy reserves, fluids, or any other consumable resource) are also necessarily limited. A wide range of applications illustrates these challenges, from closed-loop insulin delivery ("artificial pancreas") to the operation of autonomous drones, as well as many biological, ecological, or technical systems. The consideration of such constraints in the design of control laws is, to some extent, well established. However, in many cases, resources are only available for a fixed duration, or safety requirements impose restrictions on their use over a given time horizon. This is precisely the case for any control problem involving limited resources during a mission (insulin reservoirs, fuel, battery energy, etc.). One could refer to these as "memory constraints", since they govern the evolution and availability of resources over time. Such constraints still raise challenges that remain unexplored in the literature. The objective of this thesis is to analyze their impact and to develop a methodological framework that rigorously integrates them into the design of safety-critical control systems.

** Problematic and general context:

The study of control problems with hard integral constraints is crucial because it mirrors real-world limitations that cannot be ignored. In many practical settings⬔such as drug delivery, energy management, or resource-limited engineering systems⬔there are strict bounds on how much of a resource may be consumed over time. Classical control designs typically assume unlimited actuation, leading to solutions that may be mathematically optimal yet practically infeasible.

In this work, the resource limitation is expressed as a constraint on the total (integral) magnitude of the control input. Specifically, over a fixed time horizon, the integral of the control signal must remain below a prescribed budget. Our objective is to design control laws that guarantee stabilization under this constraint. However, the structure of such laws is still poorly characterized in the literature: it is unclear how to systematically construct stabilizing feedback under an integral bound, whether feedback solutions always exist, and under what conditions "bang-bang" or saturating behaviors emerge.

The problem of designing control laws under integral constraints has been addressed primarily in the context of optimal control. For example, in his work on pharmacokinetics [2], Bellman formulated dosage-constrained problems and demonstrated how dynamic programming could be used to enforce cumulative limits. Motivated by fuel-limited flight planning for unmanned aerial vehicles, subsequent researchers have investigated the minimum-fuel control problem, see e.g. [1, Chapters 6 and 8], which parallels integral constraints by emphasizing efficiency and resource management; such studies have employed linear-programming-based tools [5] and PDE-based approaches [3,4].

Yet these approaches do not directly answer a more practical question: how can one track or stabilize a desired trajectory while ensuring that the total available control effort is not exceeded? This gap motivates a deeper examination of control law structures and synthesis methods under hard integral constraints.

Some open problems that could be considered for the Ph.D. are: -- Under which conditions the system is controllable? If is it not controllable, can we characterize the reachable states, in particular are the steady states controllable? -- Dealing with optimal control problems. It is natural to look for a control minimizing some cost. What are the optimality conditions, can the Pontryagin maximum principle be applied in this case? -- Can we build a stabilizing feedback law that satisfies the constraints? Can it in addition minimize some cost? Can Lyapunov theory be used to find a feedback law?

** References: [1] M. Athans and P. L. Falb. Optimal control. McGraw-Hill Electr. Electron. Eng. Ser. McGraw-Hill Book Company, New York, NY, 1966. [2] R. Bellman. Topics in pharmacokinetics, III: Repeated dosage and impulse control. Math. Biosci., 12(1):15, 1971. [3] A. Kumar and A. Vladimirsky. An ecient method for multiobjective optimal control and optimal control subject to integral constraints. J. Comput. Math., 28(4):517551, 2010. [4] I. Mitchell and S. Sastry. Continuous path planning with multiple constraints. In 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475), volume 5, pages 55025507 Vol.5, 2003. [5] C. M. Waespy. An application of linear programming to minimum fuel optimal control. Technical Report 67-16, University of California, Los Angeles. NASA, June 1967.