Calculus of Variations and Optimal Control

 Faculteit Science and Engineering Jaar 2020/21 Vakcode WBMA016-05 Vaknaam Calculus of Variations and Optimal Control Niveau(s) bachelor Voertaal Engels Periode semester I b ECTS 5 Rooster rooster.rug.nl

Uitgebreide vaknaam Calculus of Variations and Optimal Control
Leerdoelen 1. The student is able to apply the reasoning of the calculus of variations to concrete examples, derive the Euler-Lagrange equations, and to derive minor extensions of the theory.
2. The student is able to solve optimal control problems through the use of the Minimum principle.
3. The student is able to apply the reasoning of dynamic programming to optimal control problems.
4. The student is able to solve linear quadratic optimal control problems.
5. The student is able to make connections between different aspects of optimal control and critically compare different approaches.
6. The student is able to investigate stability of equilibria of nonlinear systems by the use of Lyapunov’s first or second method, and is able to compare and combine these two methods.
Omschrijving This course deals with the basic theory of the calculus of variations and optimal control of dynamical systems.
The principles of the calculus of variations are treated, including the Euler-Lagrange equations from classical mechanics. Special attention is paid to classical problems such as the brachistochrone problem of Johann Bernoulli who formulated and solved this problem while being at the University of Groningen (1695-1705). For optimal control we treat two related solution methods: Pontryagin's Minimum principle and the Hamilton-Jacobi-Bellman theory of dynamic programming. The first method is firmly rooted in the calculus of variations. The second method extends the theory of mathematical programming in static optimization theory to dynamical systems. As an important special case we treat the linear-quadratic (LQ) optimal control problem, in which case the Hamilton-Jacobi-Bellman equation reduces to a Riccati differential equation. We also deal with the infinite-horizon version of the LQ problem.
The last part covers Lyapunov stability theory of nonlinear dynamical systems, including Lyapunov functions, LaSalle's Invariance principle, and linearization. Furthermore, the basic theory of stabilization is discussed together with its connections to optimal control.
Below a list of topics to be treated:
- Variations, Euler-Lagrange equation
- Beltrami identity
- Conditions for minimality
- Higher-order Euler-Lagrange equation
- Lagrange multiplier method
- The Minimum principle
- Variable final time
- Dynamic programming in discrete time
- Bellman's equation
- Riccati differential equation
- Completion of the squares for the LQ problem
- Infinite-horizon LQ problem and algebraic Riccati equation
- Relation with invariant subspaces of Hamiltonian matrices
- Definition of (asymptotic) stability
- Lyapunov's second method for stability
- LaSalles invariance principle
- Lyapunovs first method
- Stabilization
Uren per week
Onderwijsvorm Hoorcollege (LC), Opdracht (ASM), Werkcollege (T)
Toetsvorm Opdracht (AST), Schriftelijk tentamen (WE)
(Assessment takes place through homework assignments and written exam. The final grade is obtained by taking the following weighted average: the average of the homework assignments counts for 20% and the grade of the final exam counts for 80%. If the grade of the final exam is larger than this weighted average, then the final grade is equal to the grade of the final exam.)
Vaksoort bachelor
Coördinator A.M.S. Waters, PhD.
Docent(en) A.M.S. Waters, PhD.
Verplichte literatuur
Titel Auteur ISBN Prijs
Lecture Notes 'Calculus of Variations and Optimal Control' A.J. van der Schaft
Entreevoorwaarden Prior knowledge of Calculus, Linear Algebra, and Differential Equations is required. Also basic knowledge of systems and control theory is helpful.
Opmerkingen This course was registered last year with course code WIVOB-09
Opgenomen in
Opleiding Jaar Periode Type
BSc Applied Mathematics 3 semester I b verplicht
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BSc Mathematics: General Mathematics  (Electives and Minor General Mathematics) 3 semester I b keuze
BSc Mathematics: Probability and Statistics 3 semester I b keuze
MSc Industrial Engineering and Management: Production Technology and Logistics  (Optional technical PTL modules) - semester I b Elective SSCA