Calculus of Variations and Optimal Control
Faculteit  Science and Engineering 
Jaar  2017/18 
Vakcode  WIVOB09 
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 is about the basic theory of calculus of variations and optimal control of dynamical systems. The principles of the calculus of variations are treated, includ 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 optimal control problem, in which case the HamiltonJacobiBellman equation reduces to a Riccati differential equation. 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, with the value function of optimal control serving as a Lyapunov function. Below a list of topics to be treated:  Variations, EulerLagrange equation  Beltrami identity  Higherorder EulerLagrange equation  Lagrange multiplier method  The Minimum principle  Variable final time  Linear quadratic (LQ) cost problem  Dynamic programming in discrete time  Bellmans equation in continuous time  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  Lyapunovs 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  prof. dr. A.J. van der Schaft  
Docent(en)  prof. dr. A.J. van der Schaft  
Verplichte literatuur 


Entreevoorwaarden  Elementary knowledge of matrixes and vectors, and differential equations.  
Opmerkingen  
Opgenomen in 
