Math 2.0

Calculus of variations and optimal control; optimization Classical problems Problem

Zermelo's navigation problem

Builds on Time-optimal controlPontryagin's maximum principleOrdinary differential equation

Problem

A boat moves in the plane with constant speed V>0V > 0 relative to the water. The water moves with a velocity field w=(w1,w2) ⁣:ℝ2→ℝ2w = (w_1, w_2) \colon \R^2 \to \R^2 of class C1C^1 with ∣w∣<V\abs{w} < V everywhere. The captain controls the heading θ(t)\theta(t), so that the position (x(t),y(t))(x(t), y(t)) obeys the ordinary differential equations x′=Vcos⁡θ+w1(x,y),y′=Vsin⁡θ+w2(x,y).x' = V \cos \theta + w_1(x, y), \qquad y' = V \sin \theta + w_2(x, y). Given two points PP and QQ, find a heading t↦θ(t)t \mapsto \theta(t) that brings the boat from PP to QQ in the least time.

Remarks

It is a time-optimal control problem, and the Pontryagin maximum principle gives necessary conditions on optimal headings. Zermelo derived a differential equation for the optimal heading along the path, known as Zermelo's navigation formula. For a constant current ww, the optimal heading is constant and the boat moves along the straight segment from PP to QQ. One aims so that the sum of V(cos⁡θ,sin⁡θ)V(\cos\theta, \sin\theta) and ww points from PP to QQ. The condition ∣w∣<V\abs{w} < V ensures that the boat can make headway in every direction.

Reading path

To understand this page from the ground up, read these 26 pages in order. The foundations come first.

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Absolute continuity, concept, no evidence levelAbsolute continuityconceptAdmissible control, concept, no evidence levelAdmissible controlconceptBang–bang control, concept, no evidence levelBang–bang controlconceptControl system, concept, no evidence levelControl systemconceptControllability, concept, no evidence levelControllabilityconceptIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionMatrix exponential, concept, no evidence levelMatrix exponentialconceptObservability, concept, no evidence levelObservabilityconceptOptimal control, concept, no evidence levelOptimal controlconceptOrdinary differential equation, concept, no evidence levelOrdinary differential…conceptRank of a matrix, concept, no evidence levelRank of a matrixconceptVector space, definition, no evidence levelVector spacedefinitionPontryagin's maximum principle, statement, no evidence levelPontryagin's maximum …statementSquare matrix, definition, no evidence levelSquare matrixdefinitionLinear map, definition, no evidence levelLinear mapdefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionPolynomial of an endomorphism, definition, no evidence levelPolynomial of an endo…definitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionCharacteristic polynomial, definition, no evidence levelCharacteristic polyno…definitionCayley–Hamilton theorem, statement, no evidence levelCayley–Hamilton theor…statementKalman rank condition, statement, no evidence levelKalman rank conditionstatementTime-optimal control, problem, no evidence levelTime-optimal controlproblemZermelo's navigation problem, problem, no evidence levelZermelo's navigation …problem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Absolute continuity concept
  2. Admissible control concept
  3. Bang–bang control concept
  4. Control system concept
  5. Controllability concept
  6. Identity matrix object
  7. Matrix definition
  8. Matrix exponential concept
  9. Observability concept
  10. Optimal control concept
  11. Ordinary differential equation concept
  12. Rank of a matrix concept
  13. Vector space definition
  14. Pontryagin's maximum principle statement
  15. Square matrix definition
  16. Linear map definition
  17. Endomorphism definition
  18. Multilinear map definition
  19. Polynomial of an endomorphism definition
  20. Alternating multilinear map definition
  21. Determinant definition
  22. Characteristic polynomial definition
  23. Cayley–Hamilton theorem statement
  24. Kalman rank condition statement
  25. Time-optimal control problem
  26. Zermelo's navigation problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Zermelo%27s_navigation_problem
About this page
TagKEXJX0YDT4
URIm2://KEXJX0YDT4
Kindproblem
WritingW0 stub
MSC49K15, 49N90, 93C15
StatusThis copy of the site has no build data. Run m2 status KEXJX0YDT4.
AboutOptimal control
UsesTime-optimal control, Pontryagin's maximum principle, Ordinary differential equation
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