Polished and extended the literature on Pontryagin. Added literature on PTOS.

Author: hurak

lectures/cont_indir_references_2.qmd

@@ -5,15 +5,20 @@ bibliography:
     - ref_calculus_variations.bib
     - ref_calculus_variations_optimal_control.bib
     - ref_numerical_optimal_control.bib
+    - ref_time_optimal_control.bib
 csl: ieee-control-systems.csl
 ---
 
-The content of this lecture is standard and is discussed by a number of books and online resources. When preparing our own material, we took much inspiration from @liberzonCalculusVariationsOptimal2011, in particular the chapters 3 (application of calculus of variations to general problem of optimal control) and chapter 4 (Pontryagin's principle), and @kirkOptimalControlTheory2004.
+The content of this lecture is standard and is discussed by a number of books and online resources. When preparing our own material, we took much inspiration from @kirkOptimalControlTheory2004. In particular, we are trying to follow Kirk's way of relating the optimal control problem to the calculus of variations. Although not available online, the printed book is fairly afordable. We also used @liberzonCalculusVariationsOptimal2011, in particular the chapters 3 (application of calculus of variations to general problem of optimal control) and chapter 4 (Pontryagin's principle). Online version of the book is freely available.
 
-We did not discuss the proof of Pontryagin's principle and we do not even command the students to go through the proof in the book. Admittedly, it is rather challenging. But understanding the very statement of the theorem, its roots in calculus of variations, and how it removes the deficiencies of the calculus of variations will suffice for our purposes.
+Other recommendable classics are @brysonAppliedOptimalControl1975 and @athansOptimalControlIntroduction2006. The popular @lewisOptimalControl2012 is a bit less detailed when it comes to the topics of this particular chapter/lecture.
 
-The transition from the calculus of variations to the optimal control, especially when it comes to the definition of Hamiltonian, is somewhat tricky. Unfortunately, it is not discussed satisfactorily in the literature. Even @liberzonCalculusVariationsOptimal2011 leaves it as an (unsolved) exercise (3.5 and 3.6) to the student. Other major textbooks avoid the topic altogether. The only treatment can be found in the paper @sussmann300YearsOptimal1997, in particular in the section "The first fork in the road: Hamilton" on page 39. The issue is so delicate that they even propose to distinguish the two types of Hamiltonian by referring to one as control Hamiltonian. 
+We did not discuss the proof of Pontryagin's principle and we do not even command the students to go through the proof in the book. Admittedly, it is rather challenging. But if you are courageous, have a look at @liberzonCalculusVariationsOptimal2011. Understanding the very statement of the theorem, its roots in calculus of variations, and how it removes the deficiencies of the calculus of variations will suffice for our purposes.
 
-The time-optimal control for linear systems, in particular bang-bang control for a double integrator is described in section 4.4.1 and 4.4.2 in @liberzonCalculusVariationsOptimal2011. But the material is quite standard and can be found in many books and lecture notes. What is often not emphasized in textbooks, however, is the fact that without any modifications, the bang bang control is rather troublesome from an implementation viewpoint – it leads to *chattering*. A dedicated research thread has evolved, driven by the needs of hard disk drive industry, which is called *(a)proximate time-optimal control* (PTOS). Many dozens of papers can be found with this keyword in the title.
+The transition from the calculus of variations to the optimal control, especially when it comes to the definition of Hamiltonian, is somewhat tricky. Unfortunately, it is not discussed satisfactorily in the literature. Even @liberzonCalculusVariationsOptimal2011 leaves it as an (unsolved) exercise (3.5 and 3.6) to the student. Other major textbooks avoid the topic altogether. We find an exceptionally insightful treatment in the paper @sussmann300YearsOptimal1997, in particular in the section "The first fork in the road: Hamilton" on page 39.  
 
-Among numerous resources available freely online, the lecture notes [@grosNumericalOptimalControl2022, Section 12.3], and @evansIntroductionMathematicalOptimal can be recommended.
+The time-optimal control for linear systems, in particular bang-bang control for a double integrator is described in section 4.4.1 and 4.4.2 in @liberzonCalculusVariationsOptimal2011. But the material is quite standard and can be found in many other books and lecture notes. 
+
+What is often not emphasized in textbooks, however, is the fact that without any modifications, the bang bang control is rather troublesome from an implementation viewpoint – it leads to *chattering*. A dedicated research thread has evolved, driven by the needs of hard disk drive industry, which is called *(a)proximate time-optimal control* (PTOS). Many dozens of papers can be found with this keyword in the title. For instance, @workmanAdaptiveProximateTimeOptimal1987, and @paoProximateTimeoptimalControl1993.
+
+Among numerous other resources available freely online, the lecture notes [@grosNumericalOptimalControl2022, Section 12.3], and @evansIntroductionMathematicalOptimal can also be recommended.