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Lecture 3. The maximum principle In the last lecture, we learned calculus of variation (CoV). The key idea of CoV for the minimization problem min u∈U J(u) can be summarized as follows. 1) Assume u∗ is a minimizer, and choose a one-parameter variation uϵ s.t. u0 = u∗ and uϵ ∈ U for ϵ small. 2) The function ϵ 7→ J(uϵ) has a minimizer at ϵ = 0. Thus it satisfies the first and second order necessary

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/Optimal_Control/2023/Lecture3.pdf - 2025-12-19

Exercise for Optimal control – Week 1 Choose two problems to solve. Disclaimer This is not a complete solution manual. For some of the exercises, we provide only partial answers, especially those involving numerical problems. If one is willing to use the solution manual, one should judge whether the solutions are correct or wrong him/herself. Exercise 1 (Fundamental lemma of CoV). Let f be a real

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/Optimal_Control/2023/ex1-sol.pdf - 2025-12-19

Exercise for Optimal control – Week 2 Choose one problem to solve. Exercise 1 (Insect control). Let w(t) and r(t) denote, respectively, the worker and reproductive population levels in a colony of insects, e.g. wasps. At any time t, 0 ≤ t ≤ T in the season the colony can devote a fraction u(t) of its effort to enlarging the worker force and the remaining fraction u(t) to producing reproductives. T

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/Optimal_Control/2023/ex2.pdf - 2025-12-19

Exercise for Optimal control – Week 3 Choose 1.5 problems to solve. Disclaimer This is not a complete solution manual. For some of the exercises, we provide only partial answers, especially those involving numerical problems. If one is willing to use the solution manual, one should judge whether the solutions are correct or wrong by him/herself. Exercise 1. Consider a harmonic oscillator ẍ + x =

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/Optimal_Control/2023/ex3-sol.pdf - 2025-12-19

Exercise for Optimal control – Week 5 Choose one problem to solve. Exercise 1. Use tent method to derive the KKT condition (google it if you don’t know) for the nonlinear optimization problem: min f(x) subject to gi(x) ≤ 0, i = 1, · · · ,m hj(x) = 0, j = 1, · · · , l where f , gi, hj are continuously differentiable real-valued functions on Rn. Exercise 2. Find a variation of inputs uϵ near u∗ that

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/Optimal_Control/2023/ex4.pdf - 2025-12-19

Exercise for Optimal control – Week 6 Choose 1.5 problems to solve. Exercise 1. Derive the policy iteration scheme for the LQR problem min u(·) ∞∑ k=1 x⊤ k Qxk + u⊤ k Ruk with Q = Q⊤ ≥ 0 and R = R⊤ > 0 subject to: xk+1 = Axk +Buk. Assume the system is stabilizable. Start the iteration with a stabilizing policy. Run the policy iteration and value iteration on a computer for the following matrices:

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/Optimal_Control/2023/ex6.pdf - 2025-12-19

5 Lecture 5. Proof of the maximum principle 5.1 The tent method We continue with the static nonlinear optimization problem: min g0(x) subject to gi(x) ≤ 0, i = 1, · · · ,m (LM) in which {gi}mi=0 ∈ C1(Rn;R). Suppose that the problem is feasible, i.e., there exists an admissible x∗ which minimizes g0(x). Recall that we defined the following sets: Ωi = {x ∈ Rn : gi(x) ≤ 0}, i = 1, · · · ,m and for x1

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/Optimal_Control/2023/lec5.pdf - 2025-12-19

7 Lecture 7. Dynamic programming II 7.1 Policy iteration In previous lecture, we studied dynamic programming for discrete time systems based on Bellman’s principle of optimality. We studied both finite horizon cost J = φ(xN ) + N−1∑ k=1 Lk(xk, uk), uk ∈ Uk and infinite horizon cost J = ∞∑ k=1 L(xk, uk), uk ∈ U(xk). The key ingredients we obtained were the Bellman equations. For finite horizon, J∗

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/Optimal_Control/2023/lec7.pdf - 2025-12-19

Boiler Modeling Goal: To present a major industrial modeling effort (pre Modelica). Practice balance equations. To illustrate that it takes time to obtain good simple models. Rodney Bell: Nature does not willingly part with its secrets! 1. Introduction 2. Global Balance Equations 3. Steam Distribution 4. The Model 5. Simulation 6. Experiments 7. Conclusions Introduction ◮ Long term research projec

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/PhysicalModeling/Lectures/Boilerseight.pdf - 2025-12-19

L1-Introduction 2022-03-07 1 Modeling Karl Johan Åström Department of Automatic Control LTH Lund University from Physics to Languages and Software 1 Modeling ØEssential for the development of science, example: Brahe, Kepler, Newton Ø Essential element of all engineering Ø Process design and optimization Ø Insight and understanding Ø Control design and optimization Ø Implementation – The internal m

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/PhysicalModeling/Lectures/L1-Introduction-six.pdf - 2025-12-19

PowerPoint Presentation Equation and Object-oriented Modeling Modeling Course – Automatic Control Hilding Elmqvist Mogram AB and Modelon AB In collaboration with: Martin Otter, Gerhard Hippman, Andrea Neumayr, Oskar Åström Assistants: Karl Johan Åström and Oskar Åström Content • Introduction • Part 1: Equation Oriented Modeling (Modia) • structural and symbolic algorithms • DAE index reduction • e

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/PhysicalModeling/Lectures/L7-Modeling_Course_Automatic_Control_-_Elmqvist.pdf - 2025-12-19

Automotive Modeling—An Overview of Model Components Contents: 1. Introduction 2. Propulsion and powertrain dynamics 3. Braking system and wheel dynamics 4. Tire–road interaction models 5. Steering and suspension dynamics 6. Chassis dynamics 7. Experiments and model calibration 8. Summary Lecture on May 5: Mathias Strandberg from Modelon will discuss automotive modeling using Modelica and Modelon I

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/PhysicalModeling/Lectures/L9B-Automotive.pdf - 2025-12-19

Physical modeling – Power systems Physical modelling – AC Power systems OLOF SAMUELSSON, INDUSTRIAL ELECTRICAL ENGINEERING AND AUTOMATIO N E S A V MW and Mvar Outline • The electric power system • Electromagnetic transients • Phasor model at steady state – power flow • Electro-mechanical and mechanical oscillations • Dynamic phasor simulation • Linearized DAE and ODE • Modal analysis • Case study:

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/PhysicalModeling/Lectures/Physical_modeling_-_Power_systems_-_Samuelsson.pdf - 2025-12-19

Neurons and Neuroscience 1. Introduction 2. Neurobiology 3. Simple models of a single neuron 4. Systems with a few neurons 5. Silicon neurons 6. Event based control 7. Summary Introduction ◮ A major challenge ◮ Golgi staining 1885 ◮ Cajal 1911 Mapping of the neurons using Golgi staining ◮ McCulloch and Pitts 1943 ◮ Wiener 1948 Cybernetics - Control and Communication in the Animal and the Machine ◮

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/PhysicalModeling/Lectures/neuronseight.pdf - 2025-12-19

PowerPoint Presentation Optimal Control and Planning CS 285: Deep Reinforcement Learning, Decision Making, and Control Sergey Levine Class Notes 1. Homework 3 is out! • Start early, this one will take a bit longer! Today’s Lecture 1. Introduction to model-based reinforcement learning 2. What if we know the dynamics? How can we make decisions? 3. Stochastic optimization methods 4. Monte Carlo tree

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/StudyCircleDeepReinforcementLearning/CS285-Lecture10-ModelBasedPlanning_Control.pdf - 2025-12-19

PowerPoint Presentation Model-Based Reinforcement Learning CS 285: Deep Reinforcement Learning, Decision Making, and Control Sergey Levine Class Notes 1. Homework 3 is out! Due next week • Start early, this one will take a bit longer! 1. Basics of model-based RL: learn a model, use model for control • Why does naïve approach not work? • The effect of distributional shift in model-based RL 2. Uncer

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/StudyCircleDeepReinforcementLearning/CS285-Lecture11-ModelBasedRL.pdf - 2025-12-19

PowerPoint Presentation Deep RL with Q-Functions CS 285: Deep Reinforcement Learning, Decision Making, and Control Sergey Levine Class Notes 1. Homework 2 is due next Monday 2. Project proposal due 9/25, that’s today! • Remember to upload to both Gradescope and CMT (see Piazza post) Today’s Lecture 1. How we can make Q-learning work with deep networks 2. A generalized view of Q-learning algorithms

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/StudyCircleDeepReinforcementLearning/CS285-Lecture8-DeepRLwithQfunctions.pdf - 2025-12-19

PowerPoint Presentation Advanced Policy Gradients CS 285: Deep Reinforcement Learning, Decision Making, and Control Sergey Levine Class Notes 1. Homework 2 due today (11:59 pm)! • Don’t be late! 2. Homework 3 comes out this week • Start early! Q-learning takes a while to run Today’s Lecture 1. Why does policy gradient work? 2. Policy gradient is a type of policy iteration 3. Policy gradient as a c

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/StudyCircleDeepReinforcementLearning/CS285-Lecture9-AdvancedPolicyGradients.pdf - 2025-12-19

CS285 Deep Reinforcement Learning HW4: Model-Based RL Due November 4th, 11:59 pm 1 Introduction The goal of this assignment is to get experience with model-based reinforcement learning. In general, model-based reinforcement learning consists of two main parts: learning a dynamics function to model observed state transitions, and then using predictions from that model in some way to decide what to

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/StudyCircleDeepReinforcementLearning/hw4.pdf - 2025-12-19

Microsoft Word - Anvisningar för opposition 19 augusti 14 19 augusti 2014 Anvisningar för opposition på examensarbeten vid LTH Oppositionen ska vara både muntlig och skriftlig och båda delar ska godkännas av respondentens examinator. Opponenten ansvarar för att respondenten och dennes examinator, senast vid seminariet där examensarbetet presenteras, får en skriftlig version av oppositionen (opposi

https://www.control.lth.se/fileadmin/control/Education/EngineeringProgram/FRTN45/2018/Anvisningar_foer_opposition_19_augusti_14.pdf - 2025-12-19