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When analyzing the motion of falling objects, it is essential to consider not only the force of gravity but also the opposing force of air resistance. A practical example involves releasing a heavy test weight during a safety check on a ship. As the weight falls from rest, gravity accelerates it downward while air resistance exerts an upward force that increases with velocity. This dynamic interplay of forces is well described by differential equations, which provide a mathematical framework...
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A state function is a thermodynamic property that depends solely on the current state of a system, irrespective of its history or how it arrived at that state. These functions are represented by capital letters, such as U, H, and S, which stand for internal energy, enthalpy, and entropy, respectively.For instance, the value of internal energy depends on the system's state variables and remains unaffected by the process path. This means that whether the system underwent a linear process or a...
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A servo system exemplifies a second-order system, featuring a proportional controller and load elements that ensure the output position aligns with the input position. The relationship between these components is described by a second-order differential equation. Applying the Laplace transform under zero initial conditions yields the transfer function, showing how inputs are converted to outputs in the system.
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The integrating factor method provides a systematic way to solve first-order linear differential equations, especially those that cannot be handled by separation of variables. This method is particularly useful in modeling time-dependent physical systems influenced by both constant inputs and resistive forces. A common example is the motion of a car subjected to a constant engine force while experiencing air resistance proportional to its velocity.In such scenarios, Newton’s second law...
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Sensitivity of Optimal Solutions to Control Problems for Second Order Evolution Subdifferential Inclusions.

Krzysztof Bartosz1, Zdzisław Denkowski1, Piotr Kalita1

  • 1Institute of Computer Science, Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland.

Applied Mathematics and Optimization
|June 23, 2015
PubMed
Summary

This study investigates the sensitivity of optimal solutions in control problems with second-order evolution subdifferential inclusions. We establish new existence results and analyze convergence properties under perturbations for enhanced control system analysis.

Keywords:
Control problemEvolution subdifferential inclusionMultifunctionPseudomonotone and maximal monotone operatorsSensitivityThe Clarke subdifferential[Formula: see text]- and [Formula: see text]-convergences

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Area of Science:

  • Control Theory
  • Optimization
  • Mathematical Analysis

Background:

  • Optimal control problems are crucial in various scientific and engineering disciplines.
  • Second-order evolution subdifferential inclusions present complex mathematical structures.
  • Understanding solution sensitivity to perturbations is vital for robust control system design.

Purpose of the Study:

  • To investigate the sensitivity of optimal solutions for control problems governed by second-order evolution subdifferential inclusions.
  • To establish new existence results for a specific class of these inclusions.
  • To analyze the convergence of minimal values and minimizers under perturbations of state relations and cost functionals.

Main Methods:

  • Establishing a new existence result for the considered class of inclusions.
  • Utilizing the theory of sequential convergence (often denoted as Gamma-convergence).
  • Implementing Kuratowski convergence for solution sets and complementary Gamma-convergence for cost functionals.

Main Results:

  • A new existence result is established for the investigated class of subdifferential inclusions.
  • The abstract scheme for the convergence of minimal values and minimizers is recalled.
  • The conditions for the abstract scheme (Kuratowski and Gamma-convergence) are successfully implemented.

Conclusions:

  • The sensitivity of optimal solutions is analyzed for perturbed second-order evolution subdifferential inclusions.
  • The study provides a framework for understanding the convergence behavior of optimal solutions.
  • The findings contribute to the theoretical foundation of robust optimal control.