We consider a tank containing a fluid. The tank is subjected to directly controlled translations and rotations. The fluid motion is described by linearized wave equations under shallow water approximations. For irrotational flows, a new variational formulation of Saint-Venant equations is proposed. This provides a simple method to establish the equations when the tank is moving. Several control configurations are studied: one and two horizontal dimensions; tank geometries (straight and nonstraight bottom, rectangular and circular shapes), tank motions (horizontal translations with and without rotations). For each configuration, we prove that the linear approximation is steady-state controllable and provide a simple and flatness-based algorithm for computing the steering open-loop control. These algorithms rely on operational calculus. They lead to second order equations in space variables whose fundamental solutions define delay operators corresponding to convolutions with compact support kernels. For each configuration, several controllability open-problems are proposed and motivated.

In this paper the flatness [M. Fliess, J. Lévine, P. Martin, and P. Rouchon, Internat. J. Control, 61 (1995), pp. 1327--1361, M. Fliess, J. Lévine, P. Martin, and P. Rouchon, IEEE Trans. Automat. Control, 44 (1999), pp. 922--937] of heavy chain systems, i.e., trolleys carrying a fixed length heavy chain that may carry a load, is addressed in the partial derivatives equations framework. We parameterize the system trajectories by the trajectories of its free end and solve the motion planning problem, namely, steering from one state to another state. When considered as a finite set of small pendulums, these systems were shown to be flat [R. M. Murray, in Proceedings of the IFAC World Congress, San Francisco, CA, 1996, pp. 395--400]. Our study is an extension to the infinite dimensional case. Under small angle approximations, these heavy chain systems are described by a one-dimensional (1D) partial differential wave equation. Dealing with this infinite dimensional description, we show how to get the explicit parameterization of the chain trajectory using (distributed and punctual) advances and delays of its free end. This parameterization results from symbolic computations. Replacing the time derivative by the Laplace variable s yields a second order differential equation in the spatial variable where s is a parameter. Its fundamental solution is, for each point considered along the chain, an entire function of s of exponential type. Moreover, for each, we show that, thanks to the Liouville transformation, this solution satisfies, modulo explicitly computable exponentials of s, the assumptions of the Paley--Wiener theorem. This solution is, in fact, the transfer function from the flat output (the position of the free end of the system) to the whole state of the system. Using an inverse Laplace transform, we end up with an explicit motion planning formula involving both distributed and punctual advances and delays operators.

International audienceA method is presented to systematically transform a general inequality-constrained optimal control problem (OCP) into a new equality-constrained OCP by means of saturation functions. The transformed OCP can be treated more conveniently within the standard calculus of variations compared to the original constrained OCP. In detail, state constraints are substituted by saturation functions and successively constructed dynamical subsystems, which constitute a (dynamical) system extension. The dimension of the subsystems corresponds to the relative degree (or order) of the respective state constraints. These dynamical subsystems are linked to the original dynamics via algebraic coupling equations. The approach results in a new equality-constrained OCP with extended state and input vectors. An additional regularization term is used in the cost to regularize the new OCP with respect to the new inputs. The regularization term has to be successively reduced to approach the original constrained solution. The new OCP can be solved in a convenient manner, since the stationarity conditions are easily determined and exploited. An important aspect of the saturation function formulation is that the constraints cannot be violated during the numerical solution. The approach is illustrated for an extended version of the well-known Goddard problem with thrust and dynamic pressure constraints and using a collocation method for its numerical solution