Capstone Projects, Solution Verification, and Practical Methodology

This section synthesizes the complete optimal control workflow developed throughout the course. Rather than introducing new theory, it explains how calculus of variations, Pontryagin’s Minimum Principle, indirect methods, direct methods, dynamic programming, and numerical optimization fit together in solving practical engineering problems.

Course Roadmap¶

Topics include:

1. Calculus of Variations

2. Pontryagin’s Minimum Principle

3. Indirect Numerical Methods

4. Direct Shooting

5. Multiple Shooting

6. Direct Collocation

7. Pseudospectral Methods

8. Costate Estimation

9. Dynamic Programming and the HJB Equation

Indirect Solution Workflow¶

The indirect approach consists of:

1. Derive first-order optimality conditions.

2. Form state, costate, and stationarity equations.

3. Solve the resulting TPBVP.

4. Recover the optimal state, control, and costate trajectories.

5. Verify second-order conditions.

Extremals versus Optimal Solutions¶

The first-order necessary conditions generate an extremal, not necessarily an optimal solution. A candidate trajectory satisfies the necessary conditions but must still satisfy second-order sufficient conditions for local optimality.

Legendre–Clebsch Condition¶

For regular problems,

is a local minimum test. For LQR,

so

Thus positive definiteness of $R$ guarantees satisfaction of the Legendre–Clebsch condition.

Direct Methods¶

Direct methods parameterize the control (or state and control), transcribe the problem into an NLP, solve the NLP, then reconstruct trajectories.

Comparing Direct and Indirect Solutions¶

Recommended comparisons:

• state trajectories;

• control histories;

• cost values;

• costates (when available through collocation multipliers);

• convergence under mesh or polynomial refinement.

Project Methodology¶

A good capstone project should:

1. choose a tractable nonlinear system;

2. solve it by one indirect method;

3. solve it by one direct method;

4. compare both solutions;

5. explain any discrepancies;

6. discuss numerical robustness and computational cost.

Common Pitfalls¶

• overly complicated dynamics;

• bang–bang controls approximated by low-order polynomials;

• singular arcs ignored;

• insufficient discretization;

• comparing only controls while ignoring costates.

Research Perspective¶

Engineering research requires interpreting imperfect numerical results, diagnosing inconsistencies, and validating conclusions using theory rather than relying solely on software output.

Suggested Capstone Workflow¶

1. Select a manageable problem.

2. Derive optimality conditions.

3. Implement an indirect solver.

4. Implement a direct solver.

5. Verify convergence.

6. Compare trajectories and costs.

7. Interpret discrepancies.

8. Draw engineering conclusions.

Connection. The workflow becomes concrete in a fixed-time ascent problem whose costates yield an analytical bilinear tangent steering law.