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In an open-loop system, such as a basic thermostat, the poles of the transfer function influence the system's response but do not determine its stability. However, when feedback is introduced to form a closed-loop system, such as an advanced thermostat that adjusts heating based on room temperature, stability is governed by the new poles of the closed-loop transfer function.
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An optimal control problem without control costs.

Mario Lefebvre1

  • 1Department of Mathematics and Industrial Engineering, Polytechnique Montréal, C.P. 6079, Succursale Centre-ville, Montréal, H3C 3A7, Canada.

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|March 10, 2023
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Summary
This summary is machine-generated.

Researchers optimized diffusion processes by finding controls that minimize expected costs without control expenses. They solved a complex differential equation to determine the optimal control strategy.

Keywords:
diffusion processesdynamic programmingfirst-passage timepartial differential equationstochastic optimal control

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Area of Science:

  • Stochastic control theory
  • Partial differential equations
  • Applied mathematics

Background:

  • Controlling complex systems like diffusion processes is crucial in various scientific fields.
  • Minimizing expected costs in systems without direct control expenses presents a unique challenge.
  • Understanding the behavior of systems reaching specific states is vital for applications.

Purpose of the Study:

  • To determine the optimal control strategy for a two-dimensional diffusion process.
  • To minimize the expected value of a cost function without incurring control costs.
  • To derive and solve the associated Hamilton-Jacobi-Bellman equation.

Main Methods:

  • Formulating the problem as a stochastic control problem.
  • Utilizing dynamic programming to derive a non-linear second-order partial differential equation for the value function.
  • Employing the method of similarity solutions to find explicit solutions to the differential equation.

Main Results:

  • The optimal control is expressed in terms of the value function.
  • Explicit solutions were found for important particular cases of the non-linear partial differential equation.
  • The study provides a framework for solving similar optimal control problems.

Conclusions:

  • The research successfully identifies optimal control strategies for diffusion processes by solving complex differential equations.
  • The method of similarity solutions proves effective for finding explicit solutions in specific scenarios.
  • This work contributes to the theoretical understanding and practical application of stochastic control.