Discretization in Time

In this section we present a time-stepping scheme to solve the semi-discretized PDE (2.32). Much has been written about ODE solvers and in particular ODE integra-tors for discretized PDEs. We want to restrict ourselves to the first order Euler method, which is the most simple example of an ODE solver. However, arbitrary time stepping schemes can be incorporated into our C++ implementation. Again, we look at the ODE system (2.32)

My˙h(t) +Kyh(t) +F(yh(t)) = Muh(t) +Qvh(t) (2.51)

yh(0) =y⁰. (2.52)

2 Optimal Control of PDEs

The goal is to find an approximation of yh(t) for given control functions uh(t) and vh(t). In order to shorten the notation we define P(t) := Muh(t) +Qv^h(t).

Furthermore, we introduce a time grid 0 = t0 < t1 < . . . with ti := iτ, where τ denotes the stepsize of the integration scheme.

It is well known, cf. [42], that explicit methods are generally not qualified for semi-discretized PDEs due to the stiffness of the resulting matrices. Thus, we start our investigations with the implicit Euler method

(M +τ K)yh(ti+1) +τ F(y^h(ti+1)) =Myh(ti) +τ P(ti). (2.53) In order to obtain the solutionyh(ti+1) we have to solve a nonlinear equation system in each time step. However, for high dimensional systems or strongly nonlinear problems this is a difficult task. This leads to the idea to combine the advantages of the explicit- and the implicit Euler method. The semi-implicit Euler method treats the nonlinear function F in an explicit way, i.e., we obtain

(M +τ K)yh(t_i+1) =Myh(t_i) +τ F(yh(t_i)) +τ P(t_i). (2.54) Although we preserve the beneficial stability behaviour of the implicit method (under suitable conditions), we only have to solve a linear system of equations. This idea is a special case of more general operator splitting methods, cf. [56]. The stability and convergence analysis of the semi-implicit Euler method can be found in [87].

Similar to the spatial discretization the time stepping scheme also provides a discrete approximation of the time integral in the cost functional (2.24). Note that in our investigations the stepsize τ is much smaller than the sampling time T, see Remark 1.8.

Remark 2.22

By using the semi-implicit Euler method the effort for solving a nonlinear ODE system is essentially the same as for solving a linear ODE system. In Chapter 4 we will see that the advantage of the SQP method is that only linear PDEs have to be solved. In contrast to that the (reduced) Newton method requires the solution of nonlinear PDEs. If the resulting ODE system is solved by this semi-implicit scheme the advantage of the SQP method is negligible, see also Section 4.1.2.

Remark 2.23

As already mentioned, most explicit ODE solvers are not suited for large scale stiff systems. However, there exist stabilized explicit methods which are developed for the application on high dimensional semi-discretized parabolic PDEs. In Chapter 6 we will successfully use a representative of the ROCK (Orthogonal-Runge-Kutta-Chebyshev) family, cf. [56]. In this context we implement the method of mass lump-ing to apply the explicit method, cf. [42].

3 Minimal Stabilizing Horizons

In this chapter we come back to the MPC method introduced in Section 1.2. As already mentioned, the crucial parameter in the analysis of the MPC algorithm is given by the length of the optimization horizon. Especially, the smallest horizon, for which the MPC feedback stabilizes the system, is of interest from the theoretical and practical point of view. In the following this horizon is calledminimal stabilizing horizon. The aim of this chapter is to estimate this particular horizon for different partial differential equations. This chapter contains the theoretical contribution of this thesis.

Theorem 1.20 provides a possibility for estimating the minimal stabilizing horizon that guarantees a desired suboptimality estimate by using the exponential controlla-bility condition (1.34). In this chapter we aim at estimating the overshoot constant C and the decay rate σ for different classes of parabolic PDEs. The results will be used for explaining the qualitative behaviour of the horizon with regard to several parameters. Furthermore, we can employ the method to give design guidelines for the stage cost l in order to shorten the horizon. Note that some exponential esti-mates for the presented equations are known in the literature, e.g. [26]. However, the exponential constants are often not specified and the control cost is not incor-porated.

Our road map is inspired by Theorem 1.20 and is given by the following scheme:

1. Find (not necessarily optimal) controls u such that the exponential controlla-bility condition (1.34) is fulfilled.

2. Calculate C and σ for this particular control.

3. Determine the minimal horizon ¯N that guarantees the desired suboptimality degree by using (1.35)

Note that the effort for finding an appropriate control u in step 1 is quite different for various types of equations. Thus, the generality of the presented. results strongly depends on the considered PDE and the control structure. In Section 3.1 we derive the exponential constants for a class of n-dimensional semilinear heat equations with distributed control (3.1.1) and compare the results with a numerical simulation (3.1.2). Afterwards, in Section 3.2 we introduce with the well known backstepping approach, cf. [62], a possibility to obtain a stabilizing feedback for the boundary controlled one dimensional heat equation. On the basis of these results we derive the exponential constants for the Dirichlet controlled heat equation in Section 3.3 and for Neumann control in Section 3.4. The influence of an additional convection

3 Minimal Stabilizing Horizons

term in the heat equations is analysed in Section 3.5. In Section 3.6 we compare our method to determine a stabilizing horizon with an approach presented in [75]. In order to demonstrate that our road map is not only applicable to parabolic systems but also to hyperbolic PDEs in Section 3.7 we look at the example of the linear wave equation.

3.1 Distributed Control

In this section we apply the steps of our road map to a semilinear heat equation with distributed control. Using a feedback control law we derive the exponential constants for two different stage costs. Afterwards, we use these results in order to explain the numerical results from the simulation.