Proceeding as in the proof of Theorem 3. But recall that in Step 3 and Step 4 of the proof of Theorem 3. Integration by parts in 3. This completes the proof. We can summarize the results by the following: critical curves for 3. Generalization xf xi tf ti t ti tf t Figure 3. Note that we eliminated high order derivatives at the price of converting the scalar function into a vector-valued function. Since we can always do this, this is one of the reasons in fact for considering functionals of the type 3. Chapter 4 Optimal control 4. We prove the following result. Theorem 4.
Thus applying the necessity of the condition 36 Chapter 4.
Optimal control that the derivative must vanish at extremal points now simply for a function from R to R! Then from Theorem 1. We now introduce an function p in order to rewrite 4. Multiplying 4. With this choice of p, 4. Thus 4.
Cook and pour
This completes the proof of the theorem. This is known as the relative stationarity condition. It should be emphasized that Theorem 4. However, if we already know that an optimal solution exists and that there is a unique critical control, then this critical control is obviously optimal. This analogy with Hamiltonian mechanics was responsible for the original motivation of the Pontryagin minimum principle, which we state below without proof.
Let F x, u, t and f x, u be continuously differentiable functions of each of their arguments. Equation 4. What is the value of k? Optimal control From Theorem 4. From 4. Generalization to vector inputs and states 41 Example. Linear systems and the Riccati equation.
It is unclear if 4. We now prove the following.
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The uniqueness can be shown as follows. If x1 , p1 and x2 , p2 satisfy 4. Note that the optimal control has the form of a time-varying state-feedback law; see Figure 4. This brings us naturally to the notion of controllability. Controllability means that any state can be driven to any other state using an appropriate control. A controllable system. An uncontrollable system. The equation 4. The following theorem gives an important characterization of controllability. Prove that the system 4.
Controllability We will not prove this theorem. In Theorem 4. Is this control unique?
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Find an expression for the corresponding state. Find an expression for k t, T, a. So far we have considered continuous inputs and we have established necessary conditions for the existence of an optimal control. We now consider a larger class of control inputs, namely piecewise continuous functions, and in Theorem 5. Roughly speaking, the optimality principle simply says that any part of an optimal trajectory is optimal. We denote the class of piecewise continuous Rm valued functions on [ti , tf ] by U[ti , tf ]. Theorem 5. Thus the second term in 5.
From Theorem 1. From 5. See Figure 5.
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With this notation, in Theorem 5. Then the following implications hold: 1. We simply repeat the argument from part 2 for the time interval [tm , tf ]. Note that the optimal control is given in the form of a time-varying state feedback. Its good to keep in mind that not every optimal control problem is solvable.
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Finite Dimensional Linear Systems. John Wiley, Burghes and A. Gelfand and S. Calculus of Variations. Dover, Sussmann and J. Calculus of Variations with applications to physics and engineering. Optimal Control Theory. Rijksuniversiteit Groningen, You may attempt as many questions as you wish, but only your best 4 questions will count towards the final mark. All questions carry equal numbers of marks. Please write your answers in dark ink preferably black or blue only. Calculators are not allowed in this exam.
Give the definition of a norm on X. Read more. Calculus of variations and optimal control theory. A concise introduction. Calculus of variations. Calculus of variations I. The calculus of variations. Calculus of Variations and Optimal Control Theory also traces the historical development of the subject and features numerous exercises, notes and references at the end of each chapter, and suggestions for further study.
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