1. Write the definition of exp(At). Present two selected methods for computing exp(AT).

  2. Write an Input/State/Output representation of a linear dynamical system. Present a general solution of this differential equation and justify your answer.

  3. Write the definition of controllability. Present the Kalman's controllability rank condition and the Hautus test. Prove one of the mentioned controllability tests.

  4. Write the definition of an A-invariant space. Prove that the image of the Kalman controllability matrix is an A-invariant space. Provide a geometric interpretation of controllability in terms of A-invariance and the image of B.

  5. Write the definition of a reachable system. Prove that each reachable state of the state space is a linear combination of columns of the Kalman controllability matrix.

  6. Write the definition of input - output equivalent representations. Then write the definitions of: controllability canonical form, controllable canonical form, observable canonical form and discuss their fundamental properties.

  7. Write theorems on decomposition of the pair (A,B) and on decomposition of the pair (C, A). Prove one of these theorems.

  8. Explain the pole placement control problem. Closed-loop interpretation of controllability: what is the relationship between controllability and the freedom of assigning the closed-loop poles to a state feedback control system? Justify it.

  9. Write and explain the Ackermann's formula. How to apply the Ackermann's formula to implement a pole-placement control for multi-input systems?

  10. Explain the concept of system stabilizability. Write the definition of the stabilizable eigenvalues. What is the relationship between stabilizability and stabilizable eigenvalues? What is the relationship between stabilizability and controllabiliy?

  11. Write the definition of observability. Present the Kalman and the Hautus observability tests. Prove one of them.

  12. Provide the geometric interpretation of observability in terms of A-invariance and the kernel of C. Justify it.

  13. Explain the concept of an observer. Discuss the construction of the Luenberger observer. Propose a sufficient condition for existence of a Luenberger observer and justify it.

  14. Explain the concept of a compensator, including the separation principle and the certainty equivalence principle. Write down the theorem on fundamental properties of a control system with a compensator and prove it.

  15. Explain the problem of Linear Quadratic Control. What is the necessary and sufficient condition for existence of the solution? Write the solution.

  16. Write the Algebraic Riccati Equation and theorems on fundamental properties of the solutions of this equation.

  17. Provide the definition of the norm of a linear system induced by the 2-norm. Given a transfer function of the considered system, write the formula allowing to compute the 2nd induced norm of this system. Prove this formula.

  18. Explain the problem of H∞ Control. What is the necessary and sufficient condition for existence of the solution of the standard H∞ output feedback control problem? Write the solution.

  19. Pontryagin's maximum (or minimum) principle. Application to minimum energy control and to time optimal control (simple examples).

  20. Explain the idea of dynamic programming (including Bellman's principle of optimality).