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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 47798, 1698]*) (*NotebookOutlinePosition[ 48572, 1728]*) (* CellTagsIndexPosition[ 48503, 1722]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ APPROACH TO CONSTRAINT SET: SIMPLE OPTIMAL CONTROL PROBLEM \t\t\t\t by Paul Scott\ \>", "Title"], Cell["Introduction", "Section"], Cell["\<\ This notebook performs the numerical calculations proposed in a paper with \ same title. References below refere to this paper. This is a shooting \ method in which one varies the inital values of the costate until the \ constraint set is reached. Activate Part One initially. Then part two can be repeated with new initial \ costate values until the constraint set is reached.\ \>", "Text"], Cell[CellGroupData[{ Cell["Part One", "Subsubtitle"], Cell[CellGroupData[{ Cell["Declaring Variables", "Section"], Cell["Number of y state variables (called n).", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(n\ = \ 2\)], "Input"], Cell[BoxData[ \(2\)], "Output"] }, Open ]], Cell["Declaration of y state variables.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(yy\ = \ Table[\ y\_i\ , \ {\ i\ , \ 1\ , \ n\ }\ ]\)], "Input"], Cell[BoxData[ \({y\_1, y\_2}\)], "Output"] }, Open ]], Cell[TextData[ "Declaration of y- costate variables (called \[Lambda])."], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Lambda]\[Lambda]\ = \ Table[\ \ \[Lambda]\_i\ \ , \ {\ i\ , 1\ , n\ }\ \ ]\)], "Input"], Cell[BoxData[ \({\[Lambda]\_1, \[Lambda]\_2}\)], "Output"] }, Open ]], Cell["Number of z state variables (called m).", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(m\ = \ 2\)], "Input"], Cell[BoxData[ \(2\)], "Output"] }, Open ]], Cell["Declaration of z-state variables.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(zz\ = \ Table[\ \ z\_i\ \ , \ {\ i\ , 1\ , m\ }\ \ ]\)], "Input"], Cell[BoxData[ \({z\_1, z\_2}\)], "Output"] }, Open ]], Cell[TextData["Declaration of z-costate variables (called \[Xi])."], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Xi]\[Xi]\ = \ Table[\ \ \[Xi]\_i\ \ , \ {\ i\ , 1\ , m}\ \ ]\)], "Input"], Cell[BoxData[ \({\[Xi]\_1, \[Xi]\_2}\)], "Output"] }, Open ]], Cell["\<\ Declaration of control variables (called u). See paper (1.0.3) and \ (1.0.4).\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(uu\ = \ Table[\ \ u\_i\ \ , \ {\ i\ , 1\ , m\ }\ \ ]\)], "Input"], Cell[BoxData[ \({u\_1, u\_2}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Defining the differential equation and constraint", "Section"], Cell[CellGroupData[{ Cell["Definition of rhs of ODE F. See paper (1.0.1)", "Subsection"], Cell["Each component of F is defined.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f\_1\ \ = \ \ \(-y\_1\)\ \ + \ \ z\_1\)], "Input"], Cell[BoxData[ \(\(-y\_1\) + z\_1\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(f\_2\ \ = \ \ \(-y\_2\)\ \ + \ \ z\_2\)], "Input"], Cell[BoxData[ \(\(-y\_2\) + z\_2\)], "Output"] }, Open ]], Cell["The components are listed in vector F.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(F\ = \ Table[\ \ f\_i\ \ , \ {\ i\ , 1\ , n\ }\ \ ]\)], "Input"], Cell[BoxData[ \({\(-y\_1\) + z\_1, \(-y\_2\) + z\_2}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Definition of constraint function G. See paper (1.0.2)", "Subsection"], Cell["Each component of g is defined.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(g\_1\ = \ \ y\_1^2\ \ + \ \ y\_2^2\ \ - \ \((z\_1^2\ \ + \ \ z\_2^2\ )\)\)], "Input"], Cell[BoxData[ \(y\_1\%2 + y\_2\%2 - z\_1\%2 - z\_2\%2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(g\_2\ = \ \ \ y\_1\ \ - \ \ z\_1\)], "Input"], Cell[BoxData[ \(y\_1 - z\_1\)], "Output"] }, Open ]], Cell["The components are listed in a vector G.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(G\ = \ Table[\ \ g\_i\ \ , \ {\ i\ , 1\ , m\ }\ \ ]\)], "Input"], Cell[BoxData[ \({y\_1\%2 + y\_2\%2 - z\_1\%2 - z\_2\%2, y\_1 - z\_1}\)], "Output"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Definition of Hamiltonian. See paper (3.0.7)", "Section"], Cell[CellGroupData[{ Cell[BoxData[ \(H\ = \ \ G\ .\ G\ \ + \ \ \[Lambda]\[Lambda]\ .\ F\ \ + \ \ \[Xi]\[Xi]\ .\ uu\)], "Input"], Cell[BoxData[ \(\((y\_1 - z\_1)\)\^2 + \((y\_1\%2 + y\_2\%2 - z\_1\%2 - z\_2\%2)\)\^2 + \((\(-y\_1\) + z\_1)\)\ \[Lambda]\_1 + \((\(-y\_2\) + z\_2)\)\ \[Lambda]\_2 + u\_1\ \[Xi]\_1 + u\_2\ \[Xi]\_2\)], "Output"] }, Open ]], Cell["\<\ The following are the partial derivatives that occur in the costate \ differential equations. This is to show them.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_\(y\_1\)\ H\)], "Input"], Cell[BoxData[ \(2\ \((y\_1 - z\_1)\) + 4\ y\_1\ \((y\_1\%2 + y\_2\%2 - z\_1\%2 - z\_2\%2)\) - \[Lambda]\_1\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_\(y\_2\)\ H\)], "Input"], Cell[BoxData[ \(4\ y\_2\ \((y\_1\%2 + y\_2\%2 - z\_1\%2 - z\_2\%2)\) - \[Lambda]\_2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_\(z\_1\)\ H\)], "Input"], Cell[BoxData[ \(\(-2\)\ \((y\_1 - z\_1)\) - 4\ z\_1\ \((y\_1\%2 + y\_2\%2 - z\_1\%2 - z\_2\%2)\) + \[Lambda]\_1\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_\(z\_2\)\ H\)], "Input"], Cell[BoxData[ \(\(-4\)\ z\_2\ \((y\_1\%2 + y\_2\%2 - z\_1\%2 - z\_2\%2)\) + \[Lambda]\_2\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["The feedback control law. See paper (3.0.10)", "Section"], Cell[BoxData[ \(Do[\ uu[\([i]\)]\ \ = \ \ \ \(-Sign[\ \ \[Xi]\[Xi][\([i]\)]\ \ ]\)\ , \ {i, 1, m}\ ]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(uu\)], "Input"], Cell[BoxData[ \({\(-Sign[\[Xi]\_1]\), \(-Sign[\[Xi]\_2]\)}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Define variables for numerical calculations", "Section"], Cell["\<\ These variables are functions of t for substitution into the differential \ equations.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(YY\ = \ Table[\ Y\_i[t]\ , \ {\ i\ , \ 1\ , \ n\ }\ ]\)], "Input"], Cell[BoxData[ \({Y\_1[t], Y\_2[t]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[CapitalGamma]\[CapitalGamma]\ = \ Table[\ \ \[CapitalGamma]\_i[t]\ \ , \ {\ i\ , 1\ , n\ }\ \ ]\)], "Input"], Cell[BoxData[ \({\[CapitalGamma]\_1[t], \[CapitalGamma]\_2[t]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ZZ\ = \ Table[\ Z\_i[t]\ \ , \ {\ i\ , \ 1\ , m\ }\ \ ]\)], "Input"], Cell[BoxData[ \({Z\_1[t], Z\_2[t]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[CapitalDelta]\[CapitalDelta]\ = \ Table[\ \ \[CapitalDelta]\_i[t]\ \ , \ {\ i\ , 1\ , m\ }\ \ ]\)], "Input"], Cell[BoxData[ \({\[CapitalDelta]\_1[t], \[CapitalDelta]\_2[t]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(UU\ = \ Table[\ \ U\_i[t]\ \ , \ {\ i\ , 1\ , m\ }\ \ ]\)], "Input"], Cell[BoxData[ \({U\_1[t], U\_2[t]}\)], "Output"] }, Open ]], Cell["\<\ The following list called \"sset\" is the list of unknown functions to be \ found by solving the differential equations set up below. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(sset\ \ = \ \ Union[\ YY\ , \ ZZ\ , \ \[CapitalGamma]\[CapitalGamma]\ , \ \[CapitalDelta]\[CapitalDelta]\ ]\)], "Input"], Cell[BoxData[ \({Y\_1[t], Y\_2[t], Z\_1[t], Z\_2[t], \[CapitalGamma]\_1[t], \[CapitalGamma]\_2[t], \[CapitalDelta]\_1[t], \[CapitalDelta]\_2[t]}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Substitution lists to insert numerical variables", "Section"], Cell[CellGroupData[{ Cell[BoxData[ \(suby\ = \ Table[\ yy[\([i]\)]\ -> \ YY[\([i]\)]\ \ , \ {i, 1, n}\ ]\)], "Input"], Cell[BoxData[ \({y\_1 \[Rule] Y\_1[t], y\_2 \[Rule] Y\_2[t]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(sub\[Lambda]\ = \ Table[\ \[Lambda]\[Lambda][\([i]\)]\ -> \ \[CapitalGamma]\[CapitalGamma][\([i]\)]\ \ , \ {i, 1, n}\ ]\)], "Input"], Cell[BoxData[ \({\[Lambda]\_1 \[Rule] \[CapitalGamma]\_1[t], \[Lambda]\_2 \[Rule] \[CapitalGamma]\_2[t]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(subz\ = \ Table[\ zz[\([i]\)]\ -> \ ZZ[\([i]\)]\ \ , \ {i, 1, m}\ ]\)], "Input"], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(subz\)\" is similar to \ existing symbol \"\!\(suby\)\"."\)], "Message"], Cell[BoxData[ \({z\_1 \[Rule] Z\_1[t], z\_2 \[Rule] Z\_2[t]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(sub\[Xi]\ = \ Table[\ \[Xi]\[Xi][\([i]\)]\ -> \ \[CapitalDelta]\[CapitalDelta][\([i]\)]\ \ , \ {i, 1, m}\ ]\)], "Input"], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(sub\\[Xi]\)\" is \ similar to existing symbol \"\!\(sub\\[Lambda]\)\"."\)], "Message"], Cell[BoxData[ \({\[Xi]\_1 \[Rule] \[CapitalDelta]\_1[t], \[Xi]\_2 \[Rule] \[CapitalDelta]\_2[t]}\)], "Output"] }, Open ]], Cell["\<\ The following list called \"sub\" is the union of all the above substitution \ lists.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(sub\ = \ Union[\ \ suby\ , \ subz\ , \ sub\[Lambda]\ , \ sub\[Xi]\ \ \ ]\)], "Input"], Cell[BoxData[ \({y\_1 \[Rule] Y\_1[t], y\_2 \[Rule] Y\_2[t], z\_1 \[Rule] Z\_1[t], z\_2 \[Rule] Z\_2[t], \[Lambda]\_1 \[Rule] \[CapitalGamma]\_1[t], \[Lambda]\_2 \[Rule] \[CapitalGamma]\_2[t], \[Xi]\_1 \[Rule] \[CapitalDelta]\_1[t], \[Xi]\_2 \[Rule] \[CapitalDelta]\_2[t]}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["For graphic analysis latter", "Section"], Cell["\<\ The variable \"GG\" is the constraint computed along the trajectories, and \ should go to zero as the trasjectory approaches the constraint set.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(GG\ \ = \ \ G\ /. \ sub\)], "Input"], Cell[BoxData[ \({Y\_1[t]\^2 + Y\_2[t]\^2 - Z\_1[t]\^2 - Z\_2[t]\^2, Y\_1[t] - Z\_1[t]} \)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Substitute numerical variables into the feedback law", "Section"], Cell[TextData[ "This substitution replaces the variable \[Xi]\[Xi] in the feedback by the \ corresponding function \[CapitalDelta]\[CapitalDelta][t] for numerical \ solution."], "Text"], Cell[BoxData[ \(Do[\ UU[\([i]\)]\ \ = \ \ \ uu[\([i]\)]\ /. \ \[Xi]\[Xi][\([i]\)]\ -> \ \[CapitalDelta]\[CapitalDelta][\([i]\)]\ \ , \ \ \ \ \ \ \ \ \ \ \ \ {i, 1, m} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(UU\)], "Input"], Cell[BoxData[ \({\(-Sign[\[CapitalDelta]\_1[t]]\), \(-Sign[\[CapitalDelta]\_2[t]]\)}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Set initial values for y-state and z-state.", "Section"], Cell[CellGroupData[{ Cell[BoxData[ \(initialy\ = \ {\ \ \ 2\ , \ 0\ }\)], "Input"], Cell[BoxData[ \({2, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(initialz\ = \ {\ 0\ , \ 1\ }\)], "Input"], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(initialz\)\" is similar \ to existing symbol \"\!\(initialy\)\"."\)], "Message"], Cell[BoxData[ \({0, 1}\)], "Output"] }, Open ]] }, Open ]], Cell["Set up state equations for y and z", "Section"], Cell["\<\ Below ystate is the set of differential equations for y, see paper (1.0.1) \ and initial conditions. The command makes a table with each entry containing \ tow parts. The first part is the DE in which the Derivative of y is set \ equal to F and the numerical variables are substituted in. The second part \ sets the initial value of y.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ystate\ = \ Table[\ {\ D[\ YY[\([i]\)]\ \ , \ t\ ]\ == \ F[\([i]\)]\ \ /. \ sub\ \ , \ \n \t\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \((\ \ YY[\([i]\)]\ /. \ t\ -> \ 0\ \ )\)\ == \ initialy[\([i]\)]\ }\ , \n \t\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\ i\ , \ 1\ , \ n\ } \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ]\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SubsuperscriptBox["Y", "1", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", \(\(-Y\_1[t]\) + Z\_1[t]\)}], ",", \(Y\_1[0] == 2\)}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SubsuperscriptBox["Y", "2", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", \(\(-Y\_2[t]\) + Z\_2[t]\)}], ",", \(Y\_2[0] == 0\)}], "}"}]}], "}"}]], "Output"] }, Open ]], Cell["\<\ Below zstate is the set of differential equations for z, see paper (1.0.4) \ and initial conditions.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(zstate\ = \ Table[\ {\ D[\ ZZ[\([i]\)]\ \ , \ t\ ]\ == \ UU[\([i]\)]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , \ \n \t\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \((\ \ ZZ[\([i]\)]\ /. \ t\ -> \ 0\ \ )\)\ == \ initialz[\([i]\)]\ }\ , \n \t\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\ i\ , \ 1 \ , \ m\ } \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ]\)], "Input"], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(zstate\)\" is similar \ to existing symbol \"\!\(ystate\)\"."\)], "Message"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SubsuperscriptBox["Z", "1", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", \(-Sign[\[CapitalDelta]\_1[t]]\)}], ",", \(Z\_1[0] == 0\)}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SubsuperscriptBox["Z", "2", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", \(-Sign[\[CapitalDelta]\_2[t]]\)}], ",", \(Z\_2[0] == 1\)}], "}"}]}], "}"}]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Part Two", "Subsubtitle"], Cell[CellGroupData[{ Cell["Set initial costate values", "Section"], Cell["\<\ Adjust initial costate values until constraint set is approached.\ \>", "Text"], Cell[BoxData[ \(initial\[Lambda]\ = \ {\ \(-200\)\ , \ \(-300\)\ }\)], "Input", CellTags->"il"], Cell[BoxData[ \({\(-200\), \(-300\)}\)], "Output"], Cell[CellGroupData[{ Cell[BoxData[ \(initial\[Xi]\ = \ {\ \(-400\)\ , \ \(-200\)\ }\)], "Input"], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(initial\\[Xi]\)\" is \ similar to existing symbol \"\!\(initial\\[Lambda]\)\"."\)], "Message"], Cell[BoxData[ \({\(-400\), \(-200\)}\)], "Output"] }, Open ]], Cell["\<\ \"tend\" is the end time for solving the trajectories. You might want to \ change this sometimes.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(tend\ \ = \ \ 1.15\)], "Input"], Cell[BoxData[ StyleBox["1.14999999999999991`", StyleBoxAutoDelete->True, PrintPrecision->3]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(tinterval\ \ = \ \ {\ t, 0, tend\ }\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"t", ",", "0", ",", StyleBox["1.14999999999999991`", StyleBoxAutoDelete->True, PrintPrecision->3]}], "}"}]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Set the costate equations", "Section"], Cell[TextData[ "The following sets the \[Lambda]-costate equations and initial conditions. \ See paper (3.0.8)"], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Lambda]state\ = \ Table[\ {\ D[\ \[CapitalGamma]\[CapitalGamma][\([i]\)]\ \ , \ t\ ]\ == \ \(-\ \[PartialD]\_\(y\_i\)\ H\)\ \ /. \ sub\ \ , \ \n \t\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \((\ \ \[CapitalGamma]\[CapitalGamma][\([i]\)]\ /. \ t\ -> \ 0\ \ )\)\ == \ initial\[Lambda][\([i]\)]\ }\ , \n \t\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\ i\ , \ 1\ , \ n\ } \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ]\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SubsuperscriptBox["\[CapitalGamma]", "1", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", \(\(-2\)\ \((Y\_1[t] - Z\_1[t])\) - 4\ Y\_1[t]\ \((Y\_1[t]\^2 + Y\_2[t]\^2 - Z\_1[t]\^2 - Z\_2[t]\^2)\) + \[CapitalGamma]\_1[t]\)}], ",", \(\[CapitalGamma]\_1[0] == \(-200\)\)}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SubsuperscriptBox["\[CapitalGamma]", "2", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", \(\(-4\)\ Y\_2[t]\ \((Y\_1[t]\^2 + Y\_2[t]\^2 - Z\_1[t]\^2 - Z\_2[t]\^2)\) + \[CapitalGamma]\_2[t]\)}], ",", \(\[CapitalGamma]\_2[0] == \(-300\)\)}], "}"}]}], "}"}]], "Output"] }, Open ]], Cell[TextData[ "The following sets the \[Xi]-costate equations and initial conditions. See \ paper (3.0.8)"], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Xi]state\ = \ Table[\ {\ D[\ \[CapitalDelta]\[CapitalDelta][\([i]\)]\ \ , \ t\ ]\ == \ \(-\ \[PartialD]\_\(z\_i\)\ H\)\ \ /. \ sub\ \ , \ \n \t\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \((\ \ \[CapitalDelta]\[CapitalDelta][\([i]\)]\ /. \ t\ -> \ 0\ \ )\)\ == \ initial\[Xi][\([i]\)]\ }\ , \n \t\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\ i\ , \ 1\ , \ m\ } \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ]\)], "Input"], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(\\[Xi]state\)\" is \ similar to existing symbol \"\!\(\\[Lambda]state\)\"."\)], "Message"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SubsuperscriptBox["\[CapitalDelta]", "1", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", \(2\ \((Y\_1[t] - Z\_1[t])\) + 4\ Z\_1[t]\ \((Y\_1[t]\^2 + Y\_2[t]\^2 - Z\_1[t]\^2 - Z\_2[t]\^2)\) - \[CapitalGamma]\_1[t]\)}], ",", \(\[CapitalDelta]\_1[0] == \(-400\)\)}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SubsuperscriptBox["\[CapitalDelta]", "2", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", \(4\ Z\_2[t]\ \((Y\_1[t]\^2 + Y\_2[t]\^2 - Z\_1[t]\^2 - Z\_2[t]\^2)\) - \[CapitalGamma]\_2[t]\)}], ",", \(\[CapitalDelta]\_2[0] == \(-200\)\)}], "}"}]}], "}"}]], "Output"] }, Open ]] }, Open ]], Cell["\<\ Putting all the DE's and initial conditions in one list called \"state\".\ \>", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(state\ = \ Flatten[Union[\ ystate\ , \ zstate\ , \ \[Lambda]state\ , \ \[Xi]state \ ]]\)], "Input"], Cell[BoxData[ \(General::"spell" \( : \ \) "Possible spelling error: new symbol name \"\!\(state\)\" is similar to \ existing symbols \!\({ystate, zstate}\)."\)], "Message"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SubsuperscriptBox["Y", "1", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", \(\(-Y\_1[t]\) + Z\_1[t]\)}], ",", \(Y\_1[0] == 2\), ",", RowBox[{ RowBox[{ SubsuperscriptBox["Y", "2", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", \(\(-Y\_2[t]\) + Z\_2[t]\)}], ",", \(Y\_2[0] == 0\), ",", RowBox[{ RowBox[{ SubsuperscriptBox["Z", "1", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", \(-Sign[\[CapitalDelta]\_1[t]]\)}], ",", \(Z\_1[0] == 0\), ",", RowBox[{ RowBox[{ SubsuperscriptBox["Z", "2", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", \(-Sign[\[CapitalDelta]\_2[t]]\)}], ",", \(Z\_2[0] == 1\), ",", RowBox[{ RowBox[{ SubsuperscriptBox["\[CapitalGamma]", "1", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", \(\(-2\)\ \((Y\_1[t] - Z\_1[t])\) - 4\ Y\_1[t]\ \((Y\_1[t]\^2 + Y\_2[t]\^2 - Z\_1[t]\^2 - Z\_2[t]\^2)\) + \[CapitalGamma]\_1[t]\)}], ",", \(\[CapitalGamma]\_1[0] == \(-200\)\), ",", RowBox[{ RowBox[{ SubsuperscriptBox["\[CapitalGamma]", "2", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", \(\(-4\)\ Y\_2[t]\ \((Y\_1[t]\^2 + Y\_2[t]\^2 - Z\_1[t]\^2 - Z\_2[t]\^2)\) + \[CapitalGamma]\_2[t]\)}], ",", \(\[CapitalGamma]\_2[0] == \(-300\)\), ",", RowBox[{ RowBox[{ SubsuperscriptBox["\[CapitalDelta]", "1", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", \(2\ \((Y\_1[t] - Z\_1[t])\) + 4\ Z\_1[t]\ \((Y\_1[t]\^2 + Y\_2[t]\^2 - Z\_1[t]\^2 - Z\_2[t]\^2)\) - \[CapitalGamma]\_1[t]\)}], ",", \(\[CapitalDelta]\_1[0] == \(-400\)\), ",", RowBox[{ RowBox[{ SubsuperscriptBox["\[CapitalDelta]", "2", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", \(4\ Z\_2[t]\ \((Y\_1[t]\^2 + Y\_2[t]\^2 - Z\_1[t]\^2 - Z\_2[t]\^2)\) - \[CapitalGamma]\_2[t]\)}], ",", \(\[CapitalDelta]\_2[0] == \(-200\)\)}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Solving equations", "Section"], Cell["\<\ \"NDSolve\" numerically solves the differential equations and initial \ conditions in list \"state\" for all the unknown functions in list \"sset\", \ over the time interval \"tinterval\". 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