Dynamic Optimization Exercises

Dynamic Optimization Exercises#

Overview of Dynamic Optimization#

Dynamic optimization deals with optimization problems that evolve over time or space. Unlike static optimization problems where all decisions are made at once, dynamic optimization involves finding optimal strategies or trajectories over a time horizon or spatial domain.

Key Characteristics#

Dynamic optimization problems typically involve:

  1. Time-dependent variables: Variables that change over time \(x(t)\)

  2. Differential equations: Mathematical relationships describing how variables evolve

  3. Boundary conditions: Initial and/or final conditions that must be satisfied

  4. Integral objectives: Objectives that accumulate value over time

Types of Dynamic Optimization#

Optimal Control Problems:

  • Find optimal control inputs \(u(t)\) to steer a dynamic system

  • Objective: \(\min \int_0^T L(x(t), u(t), t) dt + \phi(x(T))\)

  • Subject to: \(\frac{dx}{dt} = f(x(t), u(t), t)\)

Parameter Estimation Problems:

  • Estimate unknown parameters in dynamic models from experimental data

  • Match model predictions to observed data

  • Critical for model validation and system identification

Dynamic Programming:

  • Break complex problems into simpler subproblems

  • Solve recursively using Bellman’s principle of optimality

Solution Methods#

  1. Direct Methods: Discretize the problem and solve as a large NLP

  2. Indirect Methods: Derive optimality conditions and solve boundary value problems

  3. Collocation Methods: Use polynomial approximations on finite elements

Applications#

  • Chemical process control and design

  • Robotics and trajectory planning

  • Economics and finance (dynamic programming)

  • Biomedical engineering (pharmacokinetics)

  • Energy systems optimization

Contents