## Section: Overall Objectives

### Overall Objectives

CAGE's activities take place in the field of mathematical control theory, with applications in three main directions: geometric models for vision, control of quantum mechanical systems, and control of systems with uncertain dynamics.

The relations between control theory and geometry of vision rely on the notion of sub-Riemannian structure, a geometric framework which is used to measure distances in nonholonomic contexts and which has a natural and powerful control theoretical interpretation. We recall that nonholonomicity refers to the property of a velocity constraint that cannot be recast as a state constraint. In the language of differential geometry, a sub-Riemannian structure is a (possibly rank-varying) Lie bracket generating distribution endowed with a smoothly varying norm.

Sub-Riemannian geometry, and in particular the theory of associated (hypoelliptic) diffusive processes, plays a crucial role in the neurogeometrical model of the primary visual cortex due to Petitot, Citti and Sarti, based on the functional architecture first described by Hubel and Wiesel. Such a model can be used as a powerful paradigm for bio-inspired image processing, as already illustrated in the recent literature (including by members of our team). Our contributions to this field are based not only on this approach, but also on another geometric and sub-Riemannian framework for vision, based on pattern matching in the group of diffeomorphisms. In this case admissible diffeomorphisms correspond to deformations which are generated by vector fields satisfying a set of nonholonomic constraints. A sub-Riemannian metric on the infinite-dimensional group of diffeomorphisms is induced by a length on the tangent distribution of admissible velocities. Nonholonomic constraints can be especially useful to describe distortions of sets of interconnected objects (e.g., motions of organs in medical imaging).

Control theory is one of the components of the forthcoming quantum revolution (As anticipated by the recent launch of the FET Flagship on Quantum Technologies), since manipulation of quantum mechanical systems is ubiquitous in applications such as quantum computation, quantum cryptography, and quantum sensing (in particular, imaging by nuclear magnetic resonance). The efficiency of the control action has a dramatic impact on the quality of the coherence and the robustness of the required manipulation. Minimal time constraints and interaction of time scales are important factors for characterizing the efficiency of a quantum control strategy. Time scales analysis is important for evaluation approaches based on adiabatic approximation theory, which is well-known to improve the robustness of the control strategy. CAGE works for the improvement of evaluation and design tools for efficient quantum control paradigms, especially for what concerns quantum systems evolving in infinite-dimensional Hilbert spaces.

Simultaneous control of a continuum of systems with slightly different dynamics is a typical problem in quantum mechanics and also a special case of the third applicative axis to which CAGE is contributing: control of systems with uncertain dynamics. The slightly different dynamics can indeed be seen as uncertainties in the system to be controlled, and simultaneous control rephrased in terms of a robustness task. Robustification, i.e., offsetting uncertainties by suitably designing the control strategy, is a widespread task in automatic control theory, showing up in many applicative domains such as electric circuits or aerospace motion planning. If dynamics are not only subject to static uncertainty, but may also change as time goes, the problem of controlling the system can be recast within the theory of switched and hybrid systems, both in a deterministic and in a probabilistic setting. Our contributions to this research field concern both stabilization (either asymptotic or in finite time) and optimal control, where redundancies and probabilistic tools can be introduced to offset uncertainties.