Suboptimal Control of Nonlinear Object:Problem of Keeping Tabs on Reference Trajectory
An optimal control problem is formulated for a class of nonlinear systems which can be presented by system with linear structure and state-depended coefficients (SDC). The system being under the influence of uncontrollable disturbance is supposed. The linearity of the transformed system structure and the quadratic functional make it possible to pass over from the Hamilton–Jacoby–Bellman equation (HJB) to the state dependent Riccati equation (SDRE) upon the control synthesis. In thus paper the optimal control problem by nonlinear system in a task of Keeping Tabs on Reference Trajectory we decide in a key of differential game. The presented example illustrates the application of the proposed control method.
We present a method for synthesis of optimal control with feedback of nonlinear systems with separated linear part via quadratic criteria. This method is based on a specialmethod of successive approximations, which allows, under standard assumptions, to find optimal control within any finite time interval and to get the procedure of its construction. An example is provided for applying this method for synthesis of control ofa system which is similar to Watt’s centrifugal governor.
The chapter studies a dynamic risk model defined on infinite time interval, where both insurance and per-claim reinsurance policies are chosen by the insurer in order to minimize a functional of the form of variation coefficient under constraints imposed with probability one on insured's and reinsurer's risks. We show that the optimum is achieved at constant policies, the optimal reinsurance is a partial stop loss reinsurance and the optimal insurance is a combination of stop loss and deductible policies. The results are illustrated by a numerical example involving uniformly distributed claim sizes.
In this paper, we consider the classical problem of maximizing discounted utility, provided that the moment of the next purchase and receipt of a loan is random (Poisson). The purpose of the study is to take into account the uncertain waiting period for receipt of a credit in consumption decision-making. The model is formulated as the problem of optimal stochastic control. The consumer at random moments buys the product at a non-random price and at the same random moments can take and return indefinite loans. For loans, the agent continuously pays interest. He constantly receives dividends in the form of external receipt of money into the account and can accumulate non-interest non-cash money. The optimality conditions are obtained using the Lagrange multiplier method. Sufficient optimality conditions reduce to partial differential equations with variable and unknown delay. They can only be solved by using a combinations of analytic expansions with respect to a small parameter. A special difficulty is the regularization («softening») of the conditions of complementary slackness. As a result, functions were obtained that determine the optimal control of consumption purchases and the size of the loan. One can see how the consumption expenditures change as the end of the planning period approaches. First, consumption depends on money and debt not separately, but on their difference – own means of the consumer. Secondly, far from the planning horizon, consumption is small and grows as the final point in time approaches. This model can be used as part of the description of the consumer agent in dynamic stochastic general equilibrium models.
We propose a model of the Russian banking system. It is based on the problem of a macroeconomic agent ”bank” which is modelled according to the principles of aggregated description, optimality and perfect foresight. To derive the equations of the model, we use the original method of relaxation of complementary slackness conditions. The model successfully reproduces main indicators of the banking system,
This book constitutes the refereed proceedings of the 9th International Conference on Optimization and Applications, OPTIMA 2018, held in Petrovac, Montenegro, in October 2018.The 35 revised full papers and the one short paper presented were carefully reviewed and selected from 103 submissions. The papers are organized in topical sections on mathematical programming; combinatorial and discrete optimization; optimal control; optimization in economy, finance and social sciences; applications.
Book include abstracts of reports presented at the IX International Conference on Optimization Methods and Applications "Optimization and applications" (OPTIMA-2018) held in Petrovac, Montenegro, October 1 - October 5, 2018.
Proceedings include extended abstracts of reports presented at the III International Conference on Optimization Methods and Applications “Optimization and application” (OPTIMA-2012) held in Costa da Caparica, Portugal, September 23—30, 2012.
The problem of searching for optimal control of nonlinear systems is indicated. Using the algorithmic method proposed in this paper, suboptimal control of a nonlinear object is constructed. The necessary assumptions are made for using the method of extended linearization. The example demonstrates the work of an algorithmic method for synthesizing suboptimal controls, and compares the behavior of the system in two modes: with optimal and suboptimal controls.