Measure-Valued Solutions for Nonlinear Evolution Equations on Banach Spaces and Their Optimal Control

1. Front Matter

   Pages i-xiii

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2. Background Materials From Analysis

   • N. U. Ahmed, Shian Wang

   Pages 1-12

3. Measure Solutions for Deterministic Evolution Equations

   • N. U. Ahmed, Shian Wang

   Pages 13-48

4. Measure Solutions for Impulsive Systems

   • N. U. Ahmed, Shian Wang

   Pages 49-74

5. Measure Solutions for Stochastic Systems

   • N. U. Ahmed, Shian Wang

   Pages 75-102

6. Measure Solutions for Neutral Evolution Equations

   • N. U. Ahmed, Shian Wang

   Pages 103-134

7. Optimal Control of Evolution Equations

   • N. U. Ahmed, Shian Wang

   Pages 135-181

8. Examples From Physical Sciences

   • N. U. Ahmed, Shian Wang

   Pages 183-217

9. Back Matter

   Pages 219-227

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This book offers the first comprehensive presentation of measure-valued solutions for nonlinear deterministic and stochastic evolution equations on infinite dimensional Banach spaces. Unlike traditional solutions, measure-valued solutions allow for a much broader class of abstract evolution equations to be addressed, providing a broader approach.


The book presents extensive results on the existence of measure-valued solutions for differential equations that have no solutions in the usual sense. It covers a range of topics, including evolution equations with continuous/discontinuous vector fields, neutral evolution equations subject to vector measures as impulsive forces, stochastic evolution equations, and optimal control of evolution equations. The optimal control problems considered cover the existence of solutions, necessary conditions of optimality, and more, significantly complementing the existing literature.


This book will be of great interest to researchers in functional analysis, partial differential equations, dynamic systems and their optimal control, and their applications, advancing previous research and providing a foundation for further exploration of the field.

• Abstract Differential Equation
• Evolution Equation
• Stochastic Differential Equation on Hilbert Spaces
• Stochastics Differential Equation
• Stochastic Navier-Stokes Equation
• Schrodinger Equation
• Reaction Diffusion Equation
• Optimal Control of Nonstandard Problems

• University of Ottawa, Ottawa, Canada

  N. U. Ahmed

• The University of Texas at El Paso, El Paso, USA

  Shian Wang

N. U. Ahmed is an Emeritus Professor at the University of Ottawa. He has made significant contributions to functional analysis applied to systems and optimal control theory. Over his academic career, Dr. Ahmed has published 8 monographs and numerous journal articles in the general areas of systems and control theory including partial differential equations, stochastic differential equations, abstract differential equations, integral equations and functional analysis. He received the University of Ottawa George Glinski award for excellence in research in 1989. 


Shian Wang is an Assistant Professor at The University of Texas at El Paso (UTEP). Dr. Wang has published 1 monograph and a series of journal articles in the area of control systems with application to intelligent transportation systems. He received a Rising STARs Award from The University of Texas System in 2023.