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This study explores extremum problems involving control variable means, linking them to cyclic dynamical systems and finite-time thermodynamics. Optimal conditions for these cyclic modes are identified and explained.

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Area of Science:

  • Control theory
  • Dynamical systems
  • Thermodynamics

Background:

  • Extremum problems with mean values are common in control theory.
  • Understanding cyclic modes in dynamical systems is crucial for optimization.
  • Finite-time thermodynamics offers a framework for analyzing non-equilibrium processes.

Purpose of the Study:

  • To analyze extremum problems incorporating mean values of control variables.
  • To establish connections between these problems and cyclic modes of dynamical systems.
  • To derive optimality conditions for cyclic modes and link them to finite-time thermodynamics.

Main Methods:

  • Mathematical formulation of extremum problems with mean values.
  • Analysis of dynamical systems to identify and characterize cyclic modes.
  • Derivation of optimality conditions using calculus of variations or similar techniques.
  • Exploration of connections to finite-time thermodynamic principles.

Main Results:

  • Established relationships between extremum problems and cyclic dynamical system modes.
  • Derived specific optimality conditions for these cyclic modes.
  • Demonstrated the relevance of finite-time thermodynamics to these problems.

Conclusions:

  • Extremum problems with mean values are intrinsically linked to cyclic dynamical behaviors.
  • Optimality conditions for cyclic modes provide insights into system efficiency.
  • Finite-time thermodynamics provides a valuable perspective for analyzing such dynamic optimization problems.