Optimality in Optimal Control Problems

This chapter presents the conditions reiated to the optimaiity of a func-tionai. We derive the necessary conditions for optimai controi to exist and appiy them to optimai controi probiems. We aiso present sufficient conditions assuring the optimum under certain conditions. Readers are encouraged to review the iogic of the conditionai statement in Section 9.25 (p. 282). 

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Optimality in Optimal Control Problems 2. the costate equations... 

Optimality in Optimal Control Problems 3.3.1 Presence of Several Local Optima... 

Different optimal control problems can be found in the monographs and papers (Khludnev, Sokolowski, 1997 Banichuk, 1980 Barbu, 1984 Cea, 1971 Lions, 1968a, 1968b Litvinov, 1987 Mignot, 1976 Fuel, 1987 Bock, Lovisek, 1987, Haslinger et ah, 1986). 

As we know the vertical displacements of the plate defined from (2.7), (2.8) can be found as a limit of solutions to the problem (2.9)-(2.11). Two questions arise in this case. The first one is the following. Is it possible to solve an optimal control problem like (2.19) when w = w/ is defined from (2.9)-(2.11) The second question concerns relationships between solutions of (2.19) and those of the regularized optimal control problem. Our goal in this subsection is to answer these questions. 

First of all let us formulate the regularized optimal control problem. If the set F is introduced in similar way and w/ = w is found from the equation... 

In the sequel we shall study an optimal control problem. Let C (fl) be a convex, bounded and closed set. Assume that ( < 0 on T for each G. In particular, this condition provides nonemptiness for Kf. Denote the solution of (2.131) by % = introduce the cost functional... 

In the next two subsections the parameter c is supposed to be fixed. The convergence of solutions of the optimal control problem (2.134) as —> 0 will be analysed in Section 2.5.4. For this reason the -dependence of the cost functional is indicated. 

Let the set be the same as in Section 2.5.2. Consider the optimal control problem... 

Suppose that 5 is fixed for the time being. We shall prove that a solution of the optimal control problem (2.189), (2.188) exists. We choose a minimizing sequence Um U. It is bounded in and so we can assume... 

An existence theorem to the equilibrium problem of the plate is proved. A complete system of equations and inequalities fulfilled at the crack faces is found. The solvability of the optimal control problem with a cost functional characterizing an opening of the crack is established. The solution is shown to belong to the space C °° near crack points provided the crack opening is equal to zero. The results of this section are published in (Khludnev, 1996c). 

Further, in Section 3.1.4, an optimal control problem is analysed. The external forces u serve as a control. The solution existence of the optimal control problem with a cost functional describing the crack opening is proved. Finally, in Section 3.1.5, we prove C°°-regularity of the solution near crack points having a zero opening. 

In this subsection we analyse an optimal control problem. The exterior forces f,g) are chosen to minimize the cost functional... 

This functional characterizes an opening of the crack. As before, x,0 is the solution of (3.48) corresponding to f,g)- At the first step we prove the existence of the optimal control problem. The next step is to prove the C°°-regularity of the solution provided that the crack opening is zero. We fixed the parameter c in this subsection the passage to the limit, as c —> 0, is analysed in Section 3.2.4. 

Consider an approximate description of the nonpenetration condition between the crack faces which can be obtained by putting c = 0 in (3.45). Similar to the case c > 0, we can analyse the equilibrium problem of the plates and prove the solution existence of the optimal control problem of the plates with the same cost functional. We aim at the convergence proof of solutions of the optimal control problem as —> 0. In this subsection we assume that T, is a segment of a straight line parallel to the axis x. 

Bock I., Lovisek J. (1987) Optimal control problems for variational inequalities with controls in coefficients and in unilateral constraints. Apl. Mat. 32 (4), 301-314. 

An optimal control system seeks to maximize the return from a system for the minimum cost. In general terms, the optimal control problem is to find a control u which causes the system... 

If we consider the limiting case where p=0 and q O, i.e., the case where there are no unknown parameters and only some of the initial states are to be estimated, the previously outlined procedure represents a quadratically convergent method for the solution of two-point boundary value problems. Obviously in this case, we need to compute only the sensitivity matrix P(t). It can be shown that under these conditions the Gauss-Newton method is a typical quadratically convergent "shooting method." As such it can be used to solve optimal control problems using the Boundary Condition Iteration approach (Kalogerakis, 1983). 

A. Rapaport. Theory in Practice of Control and Systems, chapter Information state and guaranteed value for a class of min-max nonlinear optimal control problems, pages 391-396. World Scientific, Singapore, 1998. 

These methods are efficient for problems with initial-value ODE models without state variable and final time constraints. Here solutions have been reported that require from several dozen to several hundred model (and adjoint equation) evaluations (Jones and Finch, 1984). Moreover, any additional constraints in this problem require a search for their appropriate multiplier values (Bryson and Ho, 1975). Usually, this imposes an additional outer loop in the solution algorithm, which can easily require a prohibitive number of model evaluations, even for small systems. Consequently, control vector iteration methods are effective only when limited to the simplest optimal control problems. 

On the other hand, the optimal control problem with a discretized control profile can be treated as a nonlinear program. The earliest studies come under the heading of control vector parameterization (Rosenbrock and Storey, 1966), with a representation of U t) as a polynomial or piecewise constant function. Here the mode is solved repeatedly in an inner loop while parameters representing V t) are updated on the outside. While hill climbing algorithms were used initially, recent efficient and sophisticated optimization methods require techniques for accurate gradient calculation from the DAE model. 

As mentioned in Section IV. A, a straightforward way to deal with optimal control problems is to parameterize them as piecewise polynomial functions on a predefined set of time zones. This suboptimal representation has a number of advantages. First, the approaches developed in the previous subsection can be applied directly. Secondly, for many process control applications, control moves are actually implemented as piecewise constants on fixed time intervals, so the parameterization is adequate for this application. 

Finally, finite elements are added as decision variables in (27) not just to ensure accurate approximation (of the state and control profiles), but also to provide optimal points of discontinuity for the control profile. This dual purpose led Cuthrell and Biegler (1987) to distinguish some elements as finite-and super-elements. These roles can be combined, however, if one considers the NLP formulation of the optimal control problem given below ... 

While the reduced SQP algorithm is often suitable for parameter optimization problems, it can become inefficient for optimal control problems with many degrees of freedom (the control variables). Logsdon et al. (1990) noted this property in determining optimal reflux policies for batch distillation columns. Here, the reduced SQP method was quite successful in dealing with DAOP problems with state and control profile constraints. However, the degrees of freedom (for control variables) increase linearly with the number of elements. Consequently, if many elements are required, the effectiveness of the reduced SQP algorithm is reduced. This is due to three effects ... 

Therefore, for large optimal control problems, the efficient exploitation of the structure (to obtain 0(NE) algorithms) still remains an unsolved problem. As seen above, the structure of the problem can be complicated greatly by general inequality constraints. Moreover, the number of these constraints will also grow linearly with the number of elements. One can, in fact, formulate an infinite number of constraints for these problems to keep the profiles bounded. Of course, only a small number will be active at the optimal solution thus, adaptive constraint addition algorithms can be constructed for selecting active constraints. 

Optimal control problems have more interesting features in that control profiles are literally infinite-dimensional and attention must be paid to approximating them accurately. Here the optimality conditions can be represented implicitly by high-index DAE systems, and consequently a stable and accurate discretization is required. To demonstrate these features, the classical catalyst mixing problem of Jackson (1968) was solved with the simultaneous approach. In addition to theoretical properties of the discretization, the structure of the optimal control problem was also exploited through a chainruling strategy. 

H. J. Neusser Choosing a special pulse sequence of the dump and the pump laser pulse leads to a complete blocking of the population transfer in the CIS experiment or else makes it very efficient. We can say that a special channel is open or closed, that is, controlled by the experimental parameter. This is similar to STIRAP experiments. However, it was shown by Band and Magnes [1] that the adiabatic passage population transfer in STIRAP experiments does not represent a solution of an optimal control problem. 

The basins of attraction of the coexisting CA (strange attractor) and SC are shown in the Fig. 14 for the Poincare crosssection oyf = O.67t(mod27t) in the absence of noise [169]. The value of the maximal Lyapunov exponent for the CA is 0.0449. The presence of the control function effectively doubles the dimension of the phase space (compare (35) and (37)) and changes its geometry. In the extended phase space the attractor is connected to the basin of attraction of the stable limit cycle via an unstable invariant manifold. It is precisely the complexity of the structure of the phase space of the auxiliary Hamiltonian system (37) near the nonhyperbolic attractor that makes it difficult to solve the energy-optimal control problem. 

However, using a method proposed [60,62,95,112] for experimental analysis of the Hamiltonian flow in an extended phase space of the fluctuating system, we can exploit the analogy between the Wentzel-Freidlin and Pontryagin Hamiltonians arising in the analysis of fluctuations, and the energy-optimal control problem in a nonlinear oscillator. To see how this can be done, let us consider the fluctuational dynamics of the nonlinear oscillator (35). 

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