public DockPaneState DockState {get;}

// in order to find a dockpane you need to know it's DAML id
var pane = FrameworkApplication.DockPaneManager.Find("esri_core_ProjectDockPane");

// determine visibility
bool visible = pane.IsVisible;

// activate it
pane.Activate();

// determine dockpane state
DockPaneState state = pane.DockState;

// pin it
pane.Pin();

// hide it
pane.Hide();