This work has been written for obtaining the academic title ``Doctor of Science'' awarded by the Academy of Sciences of the Czech Republic. It contains research results achieved by the author in the years 2003-2007, mainly together with his long term collaborator Vera Zeidan from the Michigan State University (East Lansing, Michigan, USA). More specifically, the papers [62, 64, 65, 66, 67, 68, 69, 70] contribute to this work in a major way, although it is indispensable that they result from author's previous work on the discrete and time scale theories (see the list of publications of the author on pg. 135).

The primary intention of this work is to present a thorough study of the first and second order optimality conditions for variable endpoints calculus of variations and optimal control problems on time scales. These conditions include the derivation of the time scale Euler-Lagrange equation or the weak maximum principle and the transversality condition (through the first variation), and the second variation or the accessory problem. Both necessary and sufficient optimality conditions are considered for the time scale calculus of variations problem, and necessary optimality conditions are derived for the time scale control problem.

The definiteness of the quadratic functional arising as the second variation is the key concept in the second order optimality conditions. While necessary conditions are expressed in terms of the nonnegativity of the second variation, sufficient optimality conditions are phrased in terms of its coercivity or positivity. Both the nonnegativity and positivity of the second variation are also characterized by a number of equivalent conditions, including conjugate points, properties of conjoined bases of the associated Jacobi equation, and the solutions of the corresponding Riccati matrix equation.

Being a text on time scales, the presented results unify and extend the corresponding results from the continuous and discrete time theories. Apart from that, the highlights of the presented work are the following:

• We present o complete study, i.e., the necessity and sufficiency, of the calculus of variations problems on time scales with general (i.e., jointly varying) endpoints. We present sufficiency criteria either in terms of the coercivity of the second variation (traditional approach), or in terms of the positivity of the second variation (novel approach).
• The result on the equivalence between the coercivity and positivity of the quadratic functionals in the calculus of variations (Theorem 3.32) is new even for the special case of the continuous time problems with jointly varying endpoints.
• For the first time there is a rigorous study of the optimal control problems on time scales. Moreover, we consider such optimal control problems with equality control constraints and with jointly varying endpoints.
• We present a relatively simple proof of the time scale weak maximum principle (Theorem 4.16), even for control problems with jointly varying endpoints. For this we derive a ``control generalization'' of the Dubois-Reymond lemma known from the calculus of variations. To our knowledge, this proof is new even for the continuous time control problems with variable (i.e., separable or jointly varying) endpoints.
• We make a connection of the time scale control problems with the theory of time scale symplectic systems. More precisely, we show that the Jacobi systems for control problems on time scales lead naturally to time scale symplectic systems.
• We present characterizations of the nonnegativity and positivity of the quadratic functional corresponding to the time scale symplectic system without assuming any normality. These results are in terms of a natural conjoined basis (Theorems 5.10 and 5.11) or in terms of time scale explicit and implicit Riccati matrix equations (Theorems 5.19, 5.26, and 5.29). The results on implicit Riccati equations are new even for the continuous time linear Hamiltonian systems.
• We establish the embedding theorem on time scales (Theorem 2.15) without any restriction on the length of the time scale interval $[$a,b]_T, which guarantees the existence and continuous dependence of solutions of a time scale dynamic equation on the initial conditions and parameters near an already existing solution on $[$a,b]_T.

This work is divided as follows. In Chapter 1 we motivate the study of optimality conditions in variational problems and the study of problems on time scales.

In Chapter 2 we introduce the time scales calculus, its basic notions, and elementary results regarding dynamic equations on time scales. Moreover, we present as a new result the time scale embedding theorem whose technical proof is displayed in Appendix A.

The main body of this text is contained in Chapters 3-5. More precisely, Chapter 3 contains a complete study of the time scale calculus of variations problems with jointly varying endpoints. We present necessary optimality conditions as well as sufficient optimality conditions. We establish the equivalence between the coercivity and positivity of the quadratic functionals arising as the second variation in this problem. Moreover, for a subclass of problems with fixed right endpoint, we study the conjugate point theory and necessary and sufficient optimality conditions in terms of the nonexistence of such conjugate points, properties of conjoined bases of the associated Jacobi equation, and the solutions of the corresponding Riccati equation.

In Chapter 4 we develop the elements of the theory of optimal control on time scales. We consider optimal control problems with the equality control constraints and jointly varying endpoints. We define the notions of controllability and normality and establish their equivalence. We derive a ``control generalization'' of the traditional Dubois-Reymond lemma known from the calculus of variations and use it to establish the time scale weak maximum principle. We derive the first and second variations of the given time scale control problem. We apply these results to the time scale isoperimetric control problem and to another the time scale control problem (without the shift in the state variable), which turns out to be equivalent to the originally considered control problem (which does have a shift in the state variable).

In Chapter 5 we present the elements of the theory of time scale symplectic systems and the corresponding Riccati matrix equations. The aim of this chapter is more to give a taste of time scale symplectic systems and what is currently the main direction of research for the author than to present a rigorous study. But one important issue is shown in full details - this is the connection of the time scale symplectic systems with time scale control problems. This connection shows that the theory of time scale symplectic systems is the ultimate field for second order optimality conditions for such time scale variational problems.

Finally, in Appendix A we present the proof of the time scale embedding theorem (Theorem 2.15). For completeness of the presented text, we display in Appendix B the implicit and inverse function theorems known from the advanced calculus course.

Let us remark that all the results in this work remain valid if we consider complex-valued coefficients and solutions, and at the same time replace the transpose of a matrix by the conjugate transpose, and ``symmetric'' by ``Hermitian''.

At the end of of each chapter there is a section entitled ``Notes'' with comments to the literature. Moreover, in each of Chapters 3-5 we include a section on open problems and perspectives outlining possible directions of future research.

Last change: January 21, 2009. (c) Roman Simon Hilscher