02: Differential Equations & Domains Logical Foundations of Cyber-Physical Systems André Platzer Logical Foundations of Cyber-Physical Systems André Platzer André Platzer (CMU) LFCPS/02: Differential Equations & Domains LFCPS/02 1 / 18
Outline Learning Objectives 1 Introduction 2 Differential Equations 3 Examples of Differential Equations 4 Domains of Differential Equations 5 Terms First-Order Formulas Continuous Programs Summary 6 André Platzer (CMU) LFCPS/02: Differential Equations & Domains LFCPS/02 2 / 18
Outline Learning Objectives 1 Introduction 2 Differential Equations 3 Examples of Differential Equations 4 Domains of Differential Equations 5 Terms First-Order Formulas Continuous Programs Summary 6 André Platzer (CMU) LFCPS/02: Differential Equations & Domains LFCPS/02 2 / 18
Learning Objectives Differential Equations & Domains semantics of differential equations descriptive power of differential equations syntax versus semantics CT M&C CPS continuous dynamics continuous operational effects differential equations evolution domains first-order logic André Platzer (CMU) LFCPS/02: Differential Equations & Domains LFCPS/02 3 / 18
Outline Learning Objectives 1 Introduction 2 Differential Equations 3 Examples of Differential Equations 4 Domains of Differential Equations 5 Terms First-Order Formulas Continuous Programs Summary 6 André Platzer (CMU) LFCPS/02: Differential Equations & Domains LFCPS/02 3 / 18
Differential Equations as Models of Continuous Processes Example (Vector field and one solution of a differential equation) y ′ ( t ) = f ( t , y ) � � y ( t 0 ) = y 0 Intuition: André Platzer (CMU) LFCPS/02: Differential Equations & Domains LFCPS/02 4 / 18
Differential Equations as Models of Continuous Processes Example (Vector field and one solution of a differential equation) y ′ ( t ) = f ( t , y ) � � y ( t 0 ) = y 0 Intuition: At each point in space, plot the value 1 of RHS f ( t , y ) as a vector André Platzer (CMU) LFCPS/02: Differential Equations & Domains LFCPS/02 4 / 18
Differential Equations as Models of Continuous Processes Example (Vector field and one solution of a differential equation) y ′ ( t ) = f ( t , y ) � � y ( t 0 ) = y 0 Intuition: At each point in space, plot the value 1 of RHS f ( t , y ) as a vector Start at initial state y 0 at initial time t 0 2 André Platzer (CMU) LFCPS/02: Differential Equations & Domains LFCPS/02 4 / 18
Differential Equations as Models of Continuous Processes Example (Vector field and one solution of a differential equation) y ′ ( t ) = f ( t , y ) � � y ( t 0 ) = y 0 Intuition: At each point in space, plot the value 1 of RHS f ( t , y ) as a vector Start at initial state y 0 at initial time t 0 2 Follow the direction of the vector 3 André Platzer (CMU) LFCPS/02: Differential Equations & Domains LFCPS/02 4 / 18
Differential Equations as Models of Continuous Processes Example (Vector field and one solution of a differential equation) y ′ ( t ) = f ( t , y ) � � y ( t 0 ) = y 0 Intuition: At each point in space, plot the value 1 of RHS f ( t , y ) as a vector Start at initial state y 0 at initial time t 0 2 Follow the direction of the vector 3 The diagram should really show infinitely many vectors . . . André Platzer (CMU) LFCPS/02: Differential Equations & Domains LFCPS/02 4 / 18
Differential Equations as Models of Continuous Processes Example (Vector field and one solution of a differential equation) y ′ ( t ) = f ( t , y ) � � y ( t 0 ) = y 0 Intuition: At each point in space, plot the value 1 of RHS f ( t , y ) as a vector Start at initial state y 0 at initial time t 0 2 Follow the direction of the vector 3 The diagram should really show infinitely many vectors . . . x ′ = v , v ′ = a Your car’s ODE: André Platzer (CMU) LFCPS/02: Differential Equations & Domains LFCPS/02 4 / 18
Differential Equations as Models of Continuous Processes Example (Vector field and one solution of a differential equation) y ′ ( t ) = f ( t , y ) � � y ( t 0 ) = y 0 Intuition: At each point in space, plot the value 1 of RHS f ( t , y ) as a vector Start at initial state y 0 at initial time t 0 2 Follow the direction of the vector 3 The diagram should really show infinitely many vectors . . . x ′ = v , v ′ = a Well it’s a wee bit more complicated Your car’s ODE: André Platzer (CMU) LFCPS/02: Differential Equations & Domains LFCPS/02 4 / 18
Intuition for Differential Equations x 1 t 0 � x ′ ( t ) = 1 � 4 x ( t ) x ( 0 ) = 1 André Platzer (CMU) LFCPS/02: Differential Equations & Domains LFCPS/02 5 / 18
Intuition for Differential Equations x = 8 ∆ 3 1 t 0 8 � x ′ ( t ) = 1 � � x ( t +∆) := x ( t )+ 1 � 4 x ( t ) 4 x ( t )∆ � x ( 0 ) = 1 x ( 0 ) := 1 André Platzer (CMU) LFCPS/02: Differential Equations & Domains LFCPS/02 5 / 18
Intuition for Differential Equations x = 4 ∆ = 8 ∆ 3 2 1 t 0 4 8 � x ′ ( t ) = 1 � � x ( t +∆) := x ( t )+ 1 � 4 x ( t ) 4 x ( t )∆ � x ( 0 ) = 1 x ( 0 ) := 1 André Platzer (CMU) LFCPS/02: Differential Equations & Domains LFCPS/02 5 / 18
Intuition for Differential Equations ∆ = 2 x = 4 ∆ 3.375 = 8 ∆ 3 2.25 2 1.5 1 1 t 0 2 6 4 8 � x ′ ( t ) = 1 � � x ( t +∆) := x ( t )+ 1 � 4 x ( t ) 4 x ( t )∆ � x ( 0 ) = 1 x ( 0 ) := 1 André Platzer (CMU) LFCPS/02: Differential Equations & Domains LFCPS/02 5 / 18
Intuition for Differential Equations ∆ = 2 x ∆ = 1 = 4 ∆ 3.375 = 8 ∆ 3 2.25 2 1.5 1 1 t 0 2 6 1 3 5 7 4 8 � x ′ ( t ) = 1 � � x ( t +∆) := x ( t )+ 1 � 4 x ( t ) 4 x ( t )∆ � x ( 0 ) = 1 x ( 0 ) := 1 André Platzer (CMU) LFCPS/02: Differential Equations & Domains LFCPS/02 5 / 18
Intuition for Differential Equations 1 ∆ = 2 x 2 ∆ = ∆ = 1 = 4 ∆ 3.375 = 8 ∆ 3 2.25 2 1.5 1 1 t 0 2 6 1 3 5 7 4 8 � x ′ ( t ) = 1 � � x ( t +∆) := x ( t )+ 1 � 4 x ( t ) 4 x ( t )∆ � x ( 0 ) = 1 x ( 0 ) := 1 André Platzer (CMU) LFCPS/02: Differential Equations & Domains LFCPS/02 5 / 18
Intuition for Differential Equations 1 ∆ = 2 x 2 ∆ = ∆ = 1 t = 4 4 e ∆ 3.375 = 8 ∆ 3 2.25 2 1.5 1 1 t 0 2 6 1 3 5 7 4 8 � x ′ ( t ) = 1 � � x ( t +∆) := x ( t )+ 1 � 4 x ( t ) 4 x ( t )∆ � x ( 0 ) = 1 x ( 0 ) := 1 André Platzer (CMU) LFCPS/02: Differential Equations & Domains LFCPS/02 5 / 18
Outline Learning Objectives 1 Introduction 2 Differential Equations 3 Examples of Differential Equations 4 Domains of Differential Equations 5 Terms First-Order Formulas Continuous Programs Summary 6 André Platzer (CMU) LFCPS/02: Differential Equations & Domains LFCPS/02 5 / 18
The Meaning of Differential Equations What exactly is a vector field? 1 What does it mean to describe directions of evolution at every point in 2 space? Could these directions possibly contradict each other? 3 Importance of meaning The physical impacts of CPSs do not leave much room for failure. We immediately want to get into the habit of studying the behavior and exact meaning of all relevant aspects of CPS. André Platzer (CMU) LFCPS/02: Differential Equations & Domains LFCPS/02 6 / 18
Differential Equations & Initial Value Problems Definition (Ordinary Differential Equation, ODE) f : D → R n on domain D ⊆ R × R n (i.e., open connected set). Then Y : I → R n is solution of initial value problem (IVP) y ′ ( t ) = f ( t , y ) � � y ( t 0 ) = y 0 on the interval I ⊆ R , iff, for all times t ∈ I , André Platzer (CMU) LFCPS/02: Differential Equations & Domains LFCPS/02 7 / 18
Differential Equations & Initial Value Problems Definition (Ordinary Differential Equation, ODE) f : D → R n on domain D ⊆ R × R n (i.e., open connected set). Then Y : I → R n is solution of initial value problem (IVP) y ′ ( t ) = f ( t , y ) � � y ( t 0 ) = y 0 on the interval I ⊆ R , iff, for all times t ∈ I , defined ( t , Y ( t )) ∈ D 1 André Platzer (CMU) LFCPS/02: Differential Equations & Domains LFCPS/02 7 / 18
Differential Equations & Initial Value Problems Definition (Ordinary Differential Equation, ODE) f : D → R n on domain D ⊆ R × R n (i.e., open connected set). Then Y : I → R n is solution of initial value problem (IVP) y ′ ( t ) = f ( t , y ) � � y ( t 0 ) = y 0 on the interval I ⊆ R , iff, for all times t ∈ I , defined ( t , Y ( t )) ∈ D 1 time-derivative Y ′ ( t ) exists and satisfies Y ′ ( t ) = f ( t , Y ( t )) . 2 André Platzer (CMU) LFCPS/02: Differential Equations & Domains LFCPS/02 7 / 18
Differential Equations & Initial Value Problems Definition (Ordinary Differential Equation, ODE) f : D → R n on domain D ⊆ R × R n (i.e., open connected set). Then Y : I → R n is solution of initial value problem (IVP) y ′ ( t ) = f ( t , y ) � � y ( t 0 ) = y 0 on the interval I ⊆ R , iff, for all times t ∈ I , defined ( t , Y ( t )) ∈ D 1 time-derivative Y ′ ( t ) exists and satisfies Y ′ ( t ) = f ( t , Y ( t )) . 2 initial value Y ( t 0 ) = y 0 3 André Platzer (CMU) LFCPS/02: Differential Equations & Domains LFCPS/02 7 / 18
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