PART 7 Controllability , Observability

We consider a 1-D tank containing an inviscid incompressible irrotational fluid. The tank is subject to one-dimensional horizontal moves. We assume that the horizontal acceleration of the tank is small compared to the gravity constant and that the height of the fluid is small compared to the length of the tank. This motivates the use of the Saint-Venant equations [5] (also called shallow water equations) to describe the motion of the fluid; see, e.g., [2, Sec. 4.2]. After suitable scaling arguments, the length of the tank and the gravity constant can be taken to be equal to 1; see [1]. Then the dynamics equations considered are, see [3] and [1], H t (t, x) + (Hv) x (t, x) = 0, (1) v t (t, x) + H + v 2 2 x (t, x) = −u (t) , (2) v(t, 0) = v(t, 1) = 0, (3) ds dt (t) = u (t) , (4) dD dt (t) = s (t) , (5) where • H (t, x) is the height of the fluid at time t and for x ∈ [0, 1], • v (t, x) is the horizontal water velocity of the fluid in a referential attached to the tank at time t and for x ∈ [0, 1] (in the shallow water 248 PROBLEM 7.1 model, all the points on the same vertical have the same horizontal velocity), • u is the horizontal acceleration of the tank in the absolute referential, • s is the horizontal velocity of the tank, • D is the horizontal displacement of the tank. Still, by scaling arguments, we may assume that, for every steady state, H, which is then a constant function, is equal to 1; see [1]. One is interested in the local controllability of the control system Σ around the equilibrium point Of course, the total mass of the fluid is conserved so that, for every solution of (1) to (3), d dt 1 0 H (t, x) dx = 0. (6) One gets (6) by integrating (1) on [0, 1] and by using (3 together with an integration by parts.) Moreover, if H and v are of class C 1 , it follows from (2) and (3) that H x (t, 0) = H x (t, 1) (= −u (t)). 1 0 H(x)dx = 1. (10) With these notations, we can define a trajectory of the control …