Calculus of Variations and Optimal Control
Faculteit  Science and Engineering 
Jaar  2021/22 
Vakcode  WBMA01605 
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 EulerLagrange 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 EulerLagrange 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 (16951705). For optimal control we treat two related solution methods: Pontryagin's Minimum principle and the HamiltonJacobiBellman 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 linearquadratic (LQ) optimal control problem, in which case the HamiltonJacobiBellman equation reduces to a Riccati differential equation. We also deal with the infinitehorizon 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, EulerLagrange equation  Beltrami identity  Conditions for minimality  Higherorder EulerLagrange equation  Lagrange multiplier method  The Minimum principle  Variable final time  Linear quadratic (LQ) problem  Dynamic programming in discrete time  Bellman's equation  Riccati differential equation  Completion of the squares for the LQ problem  Infinitehorizon 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)
(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 


Entreevoorwaarden  Prior knowledge of Calculus, Linear Algebra, and Differential Equations is required. Also basic knowledge of systems and control theory is helpful.  
Opmerkingen  
Opgenomen in 
