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Dynamic Corporate Transformation

Vol. II, Ch. 6 · Part 2. Dynamic Enterprise Optimization · Week 10

Optimal Control of Enterprise Systems

Learning outcomes

After completing this chapter, the reader should be able to:

  1. Explain the principles of optimal control as the solution doctrine for the DEO problems of Chapter 5.
  2. Formulate enterprise optimal control problems with declared states, controls, dynamics, and value.
  3. Construct enterprise state equations and verify their control-system credentials.
  4. Define enterprise control variables with their declared constraint sets and information structures.
  5. Formulate Hamiltonian functions and read their components economically.
  6. Derive adjoint equations and interpret costates as running shadow prices of enterprise states.
  7. Apply Pontryagin's Maximum Principle to derive candidate optimal enterprise policies.
  8. Interpret transversality conditions as horizon pricing.
  9. Analyze optimal enterprise trajectories, including bang-bang and singular structures, and certify optimality where concavity permits.
  10. Implement continuous-time enterprise control models with indirect and direct numerical methods.

Reading guide

Work through the chapter in section order; the full development, proofs, and worked examples are in the book — this page indexes them and does not replace them.

  1. Introduction to Optimal Control

    Introduction to Optimal Control

  2. Enterprise Control Systems

    Enterprise Control Systems

  3. State Equations

    State Equations

  4. Control Variables

    Control Variables

  5. Hamiltonian Formulation

    Hamiltonian Formulation

  6. Adjoint Variables

    Adjoint Variables

  7. Pontryagin's Maximum Principle

    Pontryagin's Maximum Principle

  8. Transversality Conditions

    Transversality Conditions

  9. Enterprise Applications

    Enterprise Applications

  10. Numerical Solution Methods

    Numerical Solution Methods

  11. Preparation for Dynamic Programming

    Preparation for Dynamic Programming

  12. Chapter Summary

    Chapter Summary

  13. Worked Examples

    Worked Examples

  14. Exercises

    Exercises

  15. Notes and Sources

    Notes and Sources

AXIOM

This chapter is instrumented by:

Launch the module, load the chapter model, modify inputs, run the optimization, and compare against the worked examples in the book.

Exercises

12 exercises, grouped A concept checks · B mathematical · C computational · D enterprise applications. Starred (★) exercises are on the advanced track. Full solutions appear in the Instructor's Manual, Chapter 6.

A. Concept checks

  1. 6.1
    Explain, using Table (see book), why maximizing HH rather than LL repairs the myopia of Proposition 5.
  2. 6.2
    Read each condition of Table (see book) as a governance audit: what plan defect does each residual detect, and on which worked example would it fire?

B. Mathematical exercises

  1. 6.3
    Derive the current-value costate equation m˙=ρm−∇xH~\dot{\mathbf{m}} = \rho\mathbf{m} - \nabla_\x \tilde H from (see book) by the substitution m=eρtλ\mathbf{m} = e^{\rho t}\lam, and re-derive the closed form m(t)m(t) of Example (see book).
  2. 6.4
    Verify by direct differentiation that the switching time ts=6.165t_s = 6.165 of Example (see book) satisfies m(ts)=3m(t_s) = 3, and compute dts/dcdt_s/dc: how does the switch respond to the unit cost of R&D?
  3. 6.5
    For Example (see book), derive g0=2.4/(e0.8−1)g_0 = 2.4/(e^{0.8} - 1) from the budget equality, verify the discounted cost 2.9372.937, and confirm μ=0.196\mu = 0.196 equals the constant discounted marginal e−ρtc′(e∗(t))e^{-\rho t}c'(e^{*}(t)).
  4. 6.6
    In the innovation-commitment witness of Proposition (see book), complete the parameter declaration: exhibit (ℓ,c,uˉ,ρ,δ,T)(\ell, c, \bar u, \rho, \delta, T) for which both the abstention and a commitment extremal satisfy (see book), and compute both values.

C. Computational exercises

  1. 6.7
    Prove the variational-equation claim used in Proposition (see book) and Theorem (see book): under C1C^1 dynamics, the flow is differentiable in the initial state with derivative solving V˙=∇xf V\dot V = \nabla_\x f\, V, V(t0)=IV(t_0) = I, and V(t)V(t) is invertible.
  2. 6.8
    Extend Theorem (see book) to the terminal- manifold problem: with affine g\mathbf{g} and the transversality of Theorem (see book)(ii), prove Mangasarian sufficiency for mandate-constrained instances.

D. Enterprise applications

  1. 6.9
    Reproduce Example (see book)'s shooting solve: bracket λ(0)\lambda(0), locate the switch, verify J∗=12.85J^{*} = 12.85 and the pointwise maximum condition on a time grid; then re-solve by direct transcription (N=48N = 48 mesh) and reconcile the KKT multipliers with λ(t)\lambda(t).
  2. 6.10
    Implement the Riccati solve of Example (see book); reproduce K(0)=(0.406,1.090)K(0) = (0.406, 1.090) and the penalty 335335, then sweep the control weight R∈[0.1,2]R \in [0.1, 2] and plot the tracking–effort frontier.
  3. 6.11
    Formulate Meridian's covenant-management block as an EOCP (leverage state, amortization dynamics, covenant band as a state constraint) and derive the unconstrai…
  4. 6.12 ★
    Develop the constrained maximum principle for corridor- managed enterprises: state the multiplier-measure form for s(x,t)≤0s(\x, t) \le 0, prove the costate jump conditions at junctions in the first-order state-constraint case, and implement an arc-structured shooting method on the restructuring funnel of Example 5.

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All three companions consume the same seeded engine (26206), so their numbers agree by construction — the MFMF convention, carried forward.