02: Differential Equations & Domains 15-424: Foundations of - PowerPoint PPT Presentation

02: Differential Equations & Domains 15-424: Foundations of Cyber-Physical Systems Andr´ e Platzer aplatzer@cs.cmu.edu Computer Science Department Carnegie Mellon University, Pittsburgh, PA 0.5 0.4 0.3 0.2 1.0 0.1 0.8 0.6 0.4 0.2 Andr´ e Platzer (CMU) FCPS / 02: Differential Equations & Domains 1 / 12

Outline Introduction 1 Differential Equations 2 Examples of Differential Equations 3 Domains of Differential Equations 4 Andr´ e Platzer (CMU) FCPS / 02: Differential Equations & Domains 2 / 12

Outline Introduction 1 Differential Equations 2 Examples of Differential Equations 3 Domains of Differential Equations 4 Andr´ e Platzer (CMU) FCPS / 02: Differential Equations & Domains 2 / 12

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: Andr´ e Platzer (CMU) FCPS / 02: Differential Equations & Domains 3 / 12

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: 1 At each point in space, plot the value of f ( t , y ) as a vector Andr´ e Platzer (CMU) FCPS / 02: Differential Equations & Domains 3 / 12

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: 1 At each point in space, plot the value of f ( t , y ) as a vector 2 Start at initial state y 0 at initial time t 0 Andr´ e Platzer (CMU) FCPS / 02: Differential Equations & Domains 3 / 12

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: 1 At each point in space, plot the value of f ( t , y ) as a vector 2 Start at initial state y 0 at initial time t 0 3 Follow the direction of the vector Andr´ e Platzer (CMU) FCPS / 02: Differential Equations & Domains 3 / 12

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: 1 At each point in space, plot the value of f ( t , y ) as a vector 2 Start at initial state y 0 at initial time t 0 3 Follow the direction of the vector The diagram should show infinitely many vectors . . . Andr´ e Platzer (CMU) FCPS / 02: Differential Equations & Domains 3 / 12

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: 1 At each point in space, plot the value of f ( t , y ) as a vector 2 Start at initial state y 0 at initial time t 0 3 Follow the direction of the vector The diagram should show infinitely many vectors . . . x ′ = v , v ′ = a Your car’s ODE Andr´ e Platzer (CMU) FCPS / 02: Differential Equations & Domains 3 / 12

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: 1 At each point in space, plot the value of f ( t , y ) as a vector 2 Start at initial state y 0 at initial time t 0 3 Follow the direction of the vector The diagram should show infinitely many vectors . . . x ′ = v , v ′ = a Well it’s a wee bit more complicated Your car’s ODE Andr´ e Platzer (CMU) FCPS / 02: Differential Equations & Domains 3 / 12

Intuition for Differential Equations x 1 t 0 � x ′ ( t ) = 1 4 x ( t ) � x (0) = 1 Andr´ e Platzer (CMU) FCPS / 02: Differential Equations & Domains 4 / 12

Intuition for Differential Equations x ∆ = 4 2 1 t 0 4 � x ′ ( t ) = 1 4 x ( t ) � x (0) = 1 Andr´ e Platzer (CMU) FCPS / 02: Differential Equations & Domains 4 / 12

Intuition for Differential Equations x ∆ = 2 ∆ = 4 3.375 2.25 1.5 1 1 t 0 2 4 4 6 � x ′ ( t ) = 1 4 x ( t ) � x (0) = 1 Andr´ e Platzer (CMU) FCPS / 02: Differential Equations & Domains 4 / 12

Intuition for Differential Equations x ∆ = 2 4.76 ∆ = 1 3.81 ∆ = 4 3.05 2.44 1.95 1.56 1.25 1 1 t 0 1 2 2 3 4 4 4 5 6 6 � x ′ ( t ) = 1 4 x ( t ) � x (0) = 1 Andr´ e Platzer (CMU) FCPS / 02: Differential Equations & Domains 4 / 12

Intuition for Differential Equations x ∆ = 1 ∆ = 2 2 ∆ = 1 ∆ = 4 1 t 0 1 2 2 3 4 4 4 5 6 6 � x ′ ( t ) = 1 4 x ( t ) � x (0) = 1 Andr´ e Platzer (CMU) FCPS / 02: Differential Equations & Domains 4 / 12

Intuition for Differential Equations x ∆ = 1 ∆ = 2 2 ∆ = 1 t e 4 ∆ = 4 1 t 0 1 2 2 3 4 4 4 5 6 6 � x ′ ( t ) = 1 4 x ( t ) � x (0) = 1 Andr´ e Platzer (CMU) FCPS / 02: Differential Equations & Domains 4 / 12

Outline Introduction 1 Differential Equations 2 Examples of Differential Equations 3 Domains of Differential Equations 4 Andr´ e Platzer (CMU) FCPS / 02: Differential Equations & Domains 4 / 12

The Meaning of Differential Equations 1 What exactly is a vector field? 2 What does it mean to describe directions of evolution at every point in space? 3 Could directions possibly contradict each other? Importance of meaning The physical impacts of CPSs do not leave much room for failure, so we immediately want to get into the mood of consistently studying the behavior and exact meaning of all relevant aspects of CPS. Andr´ e Platzer (CMU) FCPS / 02: Differential Equations & Domains 5 / 12

Differential Equations & Initial-Value Problems Definition (Ordinary Differential Equation, ODE) f : D → R n on domain D ⊆ R × R n (i.e., open connected). Then Y : I → R n is solution of initial value problem (IVP) � y ′ ( t ) = � f ( t , y ) y ( t 0 ) = y 0 on interval I ⊆ R , iff, for all times t ∈ I , Andr´ e Platzer (CMU) FCPS / 02: Differential Equations & Domains 6 / 12

Differential Equations & Initial-Value Problems Definition (Ordinary Differential Equation, ODE) f : D → R n on domain D ⊆ R × R n (i.e., open connected). Then Y : I → R n is solution of initial value problem (IVP) � y ′ ( t ) = � f ( t , y ) y ( t 0 ) = y 0 on interval I ⊆ R , iff, for all times t ∈ I , 1 ( t , Y ( t )) ∈ D Andr´ e Platzer (CMU) FCPS / 02: Differential Equations & Domains 6 / 12

Differential Equations & Initial-Value Problems Definition (Ordinary Differential Equation, ODE) f : D → R n on domain D ⊆ R × R n (i.e., open connected). Then Y : I → R n is solution of initial value problem (IVP) � y ′ ( t ) = � f ( t , y ) y ( t 0 ) = y 0 on interval I ⊆ R , iff, for all times t ∈ I , 1 ( t , Y ( t )) ∈ D 2 Y ′ ( t ) exists and Y ′ ( t ) = f ( t , Y ( t )). Andr´ e Platzer (CMU) FCPS / 02: Differential Equations & Domains 6 / 12

Differential Equations & Initial-Value Problems Definition (Ordinary Differential Equation, ODE) f : D → R n on domain D ⊆ R × R n (i.e., open connected). Then Y : I → R n is solution of initial value problem (IVP) � y ′ ( t ) = � f ( t , y ) y ( t 0 ) = y 0 on interval I ⊆ R , iff, for all times t ∈ I , 1 ( t , Y ( t )) ∈ D 2 Y ′ ( t ) exists and Y ′ ( t ) = f ( t , Y ( t )). 3 Y ( t 0 ) = y 0 Andr´ e Platzer (CMU) FCPS / 02: Differential Equations & Domains 6 / 12

Differential Equations & Initial-Value Problems Definition (Ordinary Differential Equation, ODE) f : D → R n on domain D ⊆ R × R n (i.e., open connected). Then Y : I → R n is solution of initial value problem (IVP) � y ′ ( t ) = � f ( t , y ) y ( t 0 ) = y 0 on interval I ⊆ R , iff, for all times t ∈ I , 1 ( t , Y ( t )) ∈ D 2 Y ′ ( t ) exists and Y ′ ( t ) = f ( t , Y ( t )). 3 Y ( t 0 ) = y 0 If f ∈ C ( D , R n ), then Y ∈ C 1 ( I , R n ). Andr´ e Platzer (CMU) FCPS / 02: Differential Equations & Domains 6 / 12

Differential Equations & Initial-Value Problems Definition (Ordinary Differential Equation, ODE) f : D → R n on domain D ⊆ R × R n (i.e., open connected). Then Y : I → R n is solution of initial value problem (IVP) � y ′ ( t ) = � f ( t , y ) y ( t 0 ) = y 0 on interval I ⊆ R , iff, for all times t ∈ I , 1 ( t , Y ( t )) ∈ D 2 Y ′ ( t ) exists and Y ′ ( t ) = f ( t , Y ( t )). 3 Y ( t 0 ) = y 0 If f ∈ C ( D , R n ), then Y ∈ C 1 ( I , R n ). If f continuous, then Y continuously differentiable. Andr´ e Platzer (CMU) FCPS / 02: Differential Equations & Domains 6 / 12

Outline Introduction 1 Differential Equations 2 Examples of Differential Equations 3 Domains of Differential Equations 4 Andr´ e Platzer (CMU) FCPS / 02: Differential Equations & Domains 6 / 12

Example: A Constant Differential Equation Example (Initial value problem) � x ′ ( t ) = � 5 x (0) = 2 has a solution Andr´ e Platzer (CMU) FCPS / 02: Differential Equations & Domains 7 / 12

Example: A Constant Differential Equation Example (Initial value problem) � x ′ ( t ) = � 5 x (0) = 2 has a solution x ( t ) = 5 t + 2 Andr´ e Platzer (CMU) FCPS / 02: Differential Equations & Domains 7 / 12