Optimal control multiple solutions

Although most engineering systems have well defined unique optima, multiplicity of solutions resulting from the use of variational methods in optimal control problems may appear as shown by Luus and Cormack (1972). The uniqueness can be checked using different initial guesses, for example, for the correction factor of the Hamiltonian. 

In the present paper, the optimal solutions of the underlying optimal control problems of the Chylla Haase reactor, which have been computed by a new direct multiple shooting method, are discussed. It can be shown that the first of the two products for which physical data are given in [2] can be controlled along its required constant reaction temperature setpoint while, for the second product, this cannot be achieved because of certain mathematical and technical reasons. 

For the numerical solution of optimal control problems, there are basically two well-established approaches, the indirect approach, e. g., via the solution of multipoint boxmdary-value problems based on the necessary conditions of optimal control theory, and the direct approach via the solution of constrained nonlinear programming problems based on discretizations of the control and/or the state variables. The application of an indirect method is not advisable if the equations are too complicated or a moderate accuracy of the numerical solution is commensurate with the model accuracy. Therefore, the easier-to-handle direct approach has been chosen here. Direct collocation methods, see, e. g., Stryk [6], as well as direct multiple shooting methods, see, e. g., Bock and Plitt [1], belong to this approach. In view of forthcoming large scale problems, we will focus here on the direct multiple shooting method, since only the control variables have to be discretized for this method. This leads to lower dimensional nonlinear programming problems. 

Based on a multiple shooting method for parameter identification in differential-algebraic equations due to Heim [4], a new implementation of a direct multiple shooting method for optimal control problems has been developed, which enables the solution of problems that can be separated into different phases. In each of these phases, which might be of unknown length, the control behavior due to inequality constraints, the differential equations, even the dimensions of the state and/or the control space can differ. For the optimal control problems under investigation, the different phases are concerned with the different steps of the recipes. 

Bock, H. G., Plitt, K. J. (1984) A Multiple Shooting Algorithm for Direct Solution of Optimal Control Problems, Proc. of the 9th IFAC Worldcongress, Budapest, Hungary, Vol. IX, Colloquia 14.2, 09.2... 

OC/DO-problems can be solved very efficiently and reliably by combining the above problem discretization (piecewise control parameterization and multiple shooting state discretization) with a specifically tailored sequential quadratic programming (SQP) algorithm for the solution of the large, but structured NLP problem (2.3). Such a strategy has been implemented in the optimal control code MUSCOD [6, 13]. 

H.-G. Bock and K.-J. Plitt. A multiple shooting algorithm for direct solution of optimal control problems. Preprints of the 9th IFAC World Congress, Budapest, International Federation of Automatic Control, 1984. 

The first one is based on a classical variation method. This approach is also known as an indirect method as it focuses on obtaining the solution of the necessary conditions rather than solving the optimization directly. Solution of these conditions often results in a two-point boundary value problem (TPBVP), which is accepted that it is difficult to solve [15], Although several numerical techniques have been developed to address the solution of TPBVP, e.g. control vector iteration (CVI) and single/multiple shooting method, these methods are generally based on an iterative integration of the state and adjoint equations and are usually inefficient [16], Another difficulty relies on the fact that it requires an analytical differentiation to derive the necessary conditions. 

As noted in the introduction, a major aim of the current research is the development of "black-box" automated reactors that can produce particles with desired physicochemical properties on demand and without any user intervention. In operation, an ideal reactor would behave in the manner of Figure 12. The user would first specify the required particle properties. The reactor would then evaluate multiple reaction conditions until it eventually identified an appropriate set of reaction conditions that yield particles with the specified properties, and it would then continue to produce particles with exactly these properties until instructed to stop. There are three essential parts to any automated system—(1) physical machinery to perform the process at hand, (2) online detectors for monitoring the output of the process, and (3) decision-making software that repeatedly updates the process parameters until a product with the desired properties is obtained. The effectiveness of the automation procedure is critically dependent on the performance of these three subsystems, each of which must satisfy a number of key criteria the machinery should provide precise reproducible control of the physical process and should carry out the individual process steps as rapidly as possible to enable fast screening the online detectors should provide real-time low-noise information about the end product and the decision-making software should search for the optimal conditions in a way that is both parsimonious in terms of experimental measurements (in order to ensure a fast time-to-solution) and tolerant of noise in the experimental system. 

The above examples demonstrate the DSR concept as a useful approach to generate and interrogate simultaneously complex systems for different applications. A range of reversible reactions, in particular carbon-carbon bond-formation transformations, was used to demonstrate dynamic system formation in both organic and aqueous solutions. By applying selection pressures, the optimal constituents were subsequently selected and amplified from the dynamic system by irreversible processes under kinetic control. The DSR technique can be used not only for identification purposes, but also for evaluation of the specificities of selection pressures in one-pot processes. The nature of the selection pressure applied leads to two fundamentally different classes external selection pressures, exemplified by enzyme-catalyzed resolution, and internal selection pressures, exemplified by transformation- and/or crystallization-induced resolution. Future endeavors in this area include, for example, the exploration of more complex dynamic systems, multiple resolution schemes, and variable systemic control. 

Only some of the important works for distributed systems control shall be reviewed here. Since Butkovskii results require the explicit solution of the system equations, this restricts the results to linear systems. This drawback was removed by Katz (1964) who formulated a general maximum principle which could be applied to first order hyperbolic systems and parabolic systems without representing the system by integral equations. Lurie (1967) obtained the necessary optimality conditions using the methods of classical calculus of variations. The optimization problem was formulated as a Mayer-Bolza problem for multiple integrals. 

Well, we have picked up their challenge and present optimal solutions for their control problems. In the present paper, a direct multiple shooting... 

In this work we present examples of workflows in which immunoaffinity-purified proteins were either separated using gel electrophoresis and bands exhibiting significant change from control were analyzed, or complexed proteins were eluted from the beads, digested in solution, and analyzed. It is important to note that these protocols provide general guidelines and that several optimization steps with multiple iterations of MS will likely be required for purification of a protein complex of interest. 

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