Most of the existing theory of controllability for PDEs has been developed in the absence of **constraints on the states**. Thus, in practice, most of the available controllability results do not ensure that controlled trajectories fulfil the physical constraints of the process under consideration. Nevertheless, these constraints, often formulated as unilateral bounds on the controlled state, play a fundamental role in many applications, such as those related to **diffusion processes** (heat conduction, mathematical biology and population dynamics, etc.). In those models, which enjoy the property of **positivity preserving** of the free dynamics in the absence of control, the issue arises of whether the system under consideration can be controlled between two positive states, along a trajectory preserving the positivity property. In fact, in this setting the heat equation is one of the most paradigmatic examples since, as it is well, due to the **ill-posedness of the backward dynamics**, optimal controls for controllability problems often develop a highly oscillatory nature, thereby leading to trajectories that go beyond the physical thresholds.

Similar issues also arise in many relevant applications to **contact and multi-body dynamics in biomechanics**, modelled through differential inequalities, constraints and free boundary problems, poorly understood in control.

This issue is also relevant when validating linear PDE models, often obtained through some **asymptotic limit processes**, under suitable smallness conditions on solutions, as it is the case, for example, in **elasticity** and **fluid mechanics**. Whether controlled trajectories preserve those smallness requirements is often an open problem.

This central topic requires an in-depth investigation, starting from finite-dimensional models (some specific examples such as, for instance, the space shuttle re-entry problem have been analysed by B. Bonnard, E. Trélat and collaborators (2003), but a general theory is still lacking) to continue with the most paradigmatic PDE models (heat and wave equations). The existing theory enables **optimality systems** for a number of PDE constrained optimal control problems to be derived by suitable extensions of the classical Pontryagin Maximum Principle, but the issue of whether controllability can be achieved without saturating the constraints of the dynamics is completely open, except for some very particular cases as the 1d string equation (F. Ammar Khodja et al., Annales IHP, 2010).

**This working package will provide a new theory and corresponding algorithms to determine whether PDEs are controllable without saturating constraints, and will also lead to the optimal structure of trajectories when constraints are attained.**