Most of the existing theory of controllability for PDEs has been developed in the absence of constraints on the controls and states. Thus, in practice, available results do not guarantee that controlled trajectories fulfil the physical constraints of the processes under consideration. Nevertheless, these constraints, often formulated as unilateral bounds on the control and/or controlled state, play a fundamental role in many applications. This is particularly the case in the context of diffusion processes (heat conduction, mathematical biology and population dynamics, etc.) which enjoy the property of positivity preserving of the free dynamics, in the absence of control.
The heat equation is one of the most paradigmatic examples. Solutions remain non-negative when the initial data and applied forces are non-negative. But, can the trajectory, under the action of the control, be kept non-negative when the initial datum and final target are non-negative?
By now it is well known that optimal L^2-controls fail to fulfil this property since they develop significant oscillations that are enhanced when the time horizon of control is small. This is due to the ill-posedness of the backward parabolic dynamics and eventually leads to controlled 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 from a control theoretical viewpoint.
The topic is also relevant when validating linear PDE models, often obtained through asymptotic limit processes under suitable smallness conditions on solutions. Whether controlled trajectories preserve those smallness requirements is often an open problem.
Recent results by our team allow proving, in a number of relevant situations, including linear and semilinear heat equations, that systems can be controlled under positivity constraints on the control when the control time is long enough. Furthermore, there is a minimal time for this property to hold. In other words, constrained controllability can’t be achieved if the time horizon is too short.
Our numerical experiments also show that, often, minimal time controls develop a sparsity pattern that is not yet fully understood.