Leon Lapidus

Computational aspects of the optimal control of distributed-parameter systems

Abstract Several techniques are examined for optimal control of hyperbolic and parabolic distributed-parameter systems. These include a direct search on the performance index and a method of steepest ascent based on a maximum principle analogous to that of Pontryagin for lumped-parameter systems. Extensive numerical results suggest that for the various physical systems investigated valid solutions can be obtained.

An averaging technique for stability analysis

Abstract A technique is developed whereby the stability of autonomous or nonautonomous second order nonlinear systems can be accurately determined in a straightforward manner. The differential equations describing the system under consideration are transformed into polar coordinates and the average distance of the phase point from the point in question (usually the singular point whose stability is to be investigated) is calculated a function of time. This relationship is expressed in the form of a new set of ordinary differential equations. These equations can then be integrated to yield a prediction of the behaviour of the system in future time, by providing an approximate analytical solution to the original set of differential equations. No assumptions about small or large parameters are necessary for the application of this averaging technique. In addition to stability determination, this technique proves the existence or nonexistence of limit cycles and shows how fast a phase point approaches a singular point or a limit cycle. This information is of great importance in the design of control systems and in improving the quality of control. This averaging technique is applied to the analysis and control of a continuous stirred tank reactor (CSTR), and is used to investigate the behaviour of a CSTR under the effect of some forcing functions. It is shown small perturbations introduced into an asymptotically stable CSTR can produce large oscillations. A system comprising of a series of CSTRs is successfully analyzed with this averaging technique.

The Occurrence and Numerical Solution of Physical and Chemical Systems Having Widely Varying Time Constants

The practitioner is usually unaware of the nature of stiff systems and the associated numerical integration difficulties. Even arbitrary application of stiff methods is deemed significant enough in many areas of application to be suitable for publication. On the other hand, typical solution characteristics and the requirements on their elucidation may not be fully appreciated by the numerical analyst.

The stability of interacting populations

Abstract A general Volterra model for describing the dynamics of interacting populations is proven to be inherently stable. Conditions are given under which, for any particular application of the model, each species in the community will either oscillate about its unique equilibrium value continuously in a limit cycle or approach it asymptotically. The asymptotic approach to the steady state is proven to be exponential eventually. An estimate of the decay constant indicates faster convergence to the steady state the larger the size of the equilibrium populations, the intra-species interaction, and the ‘ nutritional value ’ of the inter-species interaction.

On the analysis of the stability of distributed reaction systems by Lyapunov's direct method

Abstract It is shown that the restriction to adiabatic perturbations in concentrating-temperature space as frequently rhade in the stability analysis of distributed reaction systems is unnecessary. By means of a carefully chosen Lyapunov functional, it is shown that non-adiabatic perturbations can be handled as easily as adiabatic ones; in fact there is no need to differentiate between the two and Lyapunovs method provides an elegant way to unify them. States which were previously shown to be only conditionally stable are shown to be completely stable. In the case of catalyst particle with arbitrary Lewis number some new results are obtained.

Singular solutions in the optimal control of lumped- and distributed-parameter systems

Abstract The necessary conditions for a singular optimal subarc in both lumped- and distributed-parameter systems are derived in this paper. Consideration is given to the synthesis of optimal trajectories when such singularities exist. Computational results are presented for a number of systems including the plug-flow tubular reactor to show that such a synthesis can be carried out. A combination of direct search and steepest ascent methods prove quite feasible as a computational tool for handling such optimal systems.

Studies in approximation methods—III: Boundary value ordinary differential equations

Abstract A numerical comparison is made between the orthogonal collocation and Runge-Kutta techniques for solving boundary value ordinary differential equations. Using various forms of the catalyst effectiveness factor system, it is shown that orthogonal collocation is the computationally superior method; this result also holds for a specific class of initial value problems. A multiple element approach in which the elements are located in an optimal manner is also developed. This optimal procedure holds promise for efficiently solving complicated boundary value problems.

Studies in approximation methods—IV: Double orthogonal collocation with an infinite domain

Abstract It is suggested that when one or more of the independent variables in a partial differential equation vary over infinite domains a single judiciously chosen orthogonal polynomial may be quite feasible. A sample problem is used to illustrate the technique and a Kronecker product representation of the constitutive equations used to simplify the development. The results of computation on this sample problem show that the overall approach permits accurate solutions at low computational cost.

Studies in approximation methods—II: Initial value ordinary differential equations

Abstract A comparison of Runge—Kutta and orthogonal collocation methods is made for the solution of initial value ordinary differential equations. The direct connection between implicit Runge—Kutta and orthogonal collocation methods is shown for a certain class of initial value problems; this class being of importance to chemical engineering systems. A number of new semi-implicit Runge—Kutta methods which have an imbedded truncation error estimate feature and are A-stable are presented. Using some test systems these new algorithms are shown to be the most computationally efficient for initial value problems.

Numerical observations on symmetric and asymmetric profiles in catalyst slabs

Abstract A numerical procedure is described for generating either symmetric or asymmetric steady-state mass and temperature profiles in a catalyst slab. Observations on the dynamic behavior of these catalyst slabs and the stability of the steady-state profiles are quoted. The validity of the frequently used assumption of symmetry or isothermality of a catalyst particle is also discussed.